IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0 


l.f 


1.25 


1.4 


IM 
1.6 


V] 


i9 


/2 


/y 


^^ 


^w 


""a 


// 


'/ 


/^, 


Photographic 

Sdences 
Corporation 


m 


v 


\\ 


■4^ 


% 


<3 


n^    #1 


o'^ 


y^ 


<?>'- 


"2  WS'JT  MAIN  O.REET 

WEBSTEM,N.Y.  U580 

(716)  873-4503 


f^^" 


CIHM/ICMH 

Microfiche 

Series. 


CIHM/ICMH 
Collection  de 
microfiches. 


Canadian  Institute  for  Historical  Microreproductions  /  Institut  canadien  de  microreproductions  historiques 


Technical  and  Bibliographic  Notes/Notes  techniques  et  bibliographiques 


The 
to  tl 


The  Institute  has  attempted  to  obtain  the  best 
original  copy  available  for  filming.  Features  of  this 
copy  which  may  be  bibliographically  unique, 
which  may  alter  any  of  the  images  in  the 
reproduction,  or  which  may  significantly  change 
the  usual  method  of  filming,  are  checked  below. 


L'Institut  a  microfilm^  le  meilleur  exemplaire 
qu'il  lui  a  6t6  possible  de  se  procurer.  Les  details 
de  cet  exemplaire  qui  sont  peut-etre  uniques  du 
point  de  vue  bibliographique,  qui  peuvent  modifier 
une  image  reproduite,  ou  qui  peuvent  exiger  une 
modification  dans  la  m^thode  normale  de  filmage 
sont  indiqu^s  ci-dessous. 


The 
pos) 
of  tl 
film 


n 


Coloured  covers/ 
Couverture  de  couleur 


□    Covers  damaged/ 
Couverture  endon 


□    Cov 
Cou 


mmagee 


Covers  restored  and/or  laminated/ 


verture  restaurde  et/ou  pellicul^e 


□    Cover  title  missing/ 
Le  titr 


tre  de  couverture  manque 


□ 


□ 


V 


Coloured  pages/ 
Pages  de  couleur 


□    Pages  damaged/ 
Pages  endommag^es 


Pages  restored  and/or  laminated/ 
Pages  restaur^es  et/ou  pelliculoes 

Pages  discoloured,  stained  or  foxed/ 
Pages  d^colorees,  tachetees  ou  piquees 


Orig 

beg 

the 

sior 

oth« 

firsi 

sior 

or  il 


□ 
□ 


Coloured  maps/ 

Cartes  g6ographiques  en  couleur 

Coloured  ink  (i.e.  other  than  blue  or  black)/ 
Encre  de  couleur  (i.e.  autre  que  bleue  ou  noire) 

Coloured  plates  and/or  illustrations/ 
Planches  et/ou  illustrations  en  couleur 

Bound  with  other  material/ 
Relie  avec  d'autres  documents 


n 


□    Tight  binding  may  cause  shadows  or  distortion 
along  interior  margin/ 

La  reliure  serree  peut  causar  de  I'ombre  ou  de  la 
distortion  le  long  de  la  marge  int^rieure 

□    Blank  leaves  added  during  restoration  may 
appear  within  the  text.  Whenever  possible,  these 
have  been  omitted  from  filming/ 
II  se  peut  que  certaines  pages  blanches  ajoutdes 
lors  d'une  restauration  apparaissent  da-^s  le  texte, 
mais,  lorsque  cela  6tait  possible,  ces  pages  n'ont 
pas  et6  filmees. 


y 


□ 
D 
D 
D 


Pages  detached/ 
Pages  detachees 

Showthrough/ 
Transparence 

Quality  of  print  varies/ 
Qualite  inegale  de  I'impression 

Includes  supplementary  material/ 
Comprend  du  materiel  supplementaire 

Only  edition  available/ 
Seule  Edition  disponible 

Pages  wholly  or  partially  obscured  by  errata 
slips,  tissues,  etc.,  have  been  refilmed  to 
ensu-e  the  best  possible  image/ 
Les  pages  totalement  ou  pardellement 
obscurcies  par  un  feuillet  d'errata,  une  pelure, 
etc.,  ont  itik  filmees  A  nouveau  de  facon  d 
obtenir  la  meilleure  image  possible. 


The 
sha 
TIN 
whi 

Mai 
diff 
enti 
beg 
righ 
reqi 
met 


□ 


Additional  comments:/ 
Commentaires  supplementaires; 


This  item  is  filmed  at  the  reduction  ratio  checked  below/ 

Ce  document  est  film^  au  taux  de  reduction  indiqu6  ci  dessous. 

10X  14X  18X  22X 


26X 


30X 


12X 


16X 


20X 


24X 


28X 


32X 


The  copy  filmed  here  has  been  reproduced  thanks 
to  the  generosity  of: 

University  of  British  Columbia  Library 


L'exemplaira  film6  fut  reproduit  grdce  d  la 
g6n6rosit6  de: 

University  of  British  Columbia  Library 


The  images  appearing  here  are  the  best  quality 
possible  considering  the  condition  and  legibility 
of  the  original  copy  and  in  keeping  with  the 
filming  contract  specifications. 


Les  images  suivantes  ont  6t6  reproduites  avec  le 
plus  grand  soin,  compte  tenu  de  la  condition  et 
de  la  nettetd  de  I'exemplaire  film6,  et  en 
conformity  svec  les  conditions  du  contrat  de 
filmage. 


Original  copies  in  printed  pap^r  covers  are  filmed 
beginning  with  the  front  cover  and  ending  on 
the  last  page  with  a  printed  or  illustrated  impres- 
sion, or  the  back  cover  when  appropriate.  All 
other  original  copies  are  filmed  beginning  on  the 
first  page  with  >>  printed  or  illustrated  impres- 
sion, and  ending  on  the  last  page  with  a  printed 
or  illustrated  impression. 


Les  exemplaires  originaux  dont  la  couverture  en 
papier  est  imprimde  sont  film6s  en  commenpant 
par  le  premier  plat  et  en  terminant  soit  par  la 
dernidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration,  soit  par  le  second 
plat,  selon  le  cas.  Tous  les  autres  exemplaires 
originaux  sont  filmds  en  commenpant  par  la 
premidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration  et  en  terminant  par 
la  dernidre  page  qui  comporte  une  telle 
empreinte. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol  — ^  (meaning   "CON- 
TINUED"), or  the  symbol  V  (meaning   "EISiD"), 
whichever  applies. 


Un  des  symboles  suivants  apparaitra  sur  la 
dernidre  image  de  chaque  microfiche,  selon  le 
cas:  le  symbole  — ^  signifie  "A  SUIVRE",  le 
symbole  V  signifie  "FIN". 


Maps,  plates,  charts,  etc.,  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  t      to  bottom,  as  many  frames  as 
required.  '   <e  following  diagrams  illustrate  the 
method: 


Les  cartes,  planches,  tableaux,  etc.,  peuvent  dtre 
filmds  d  des  taux  de  reduction  diffdrents. 
Lorsque  le  document  est  trop  grand  pour  dtre 
reproduit  en  un  seul  clichd,  il  est  filmd  d  partir 
de  Tangle  supdrieur  gauche,  de  gauche  d  droite, 
et  de  haut  en  bas,  en  prenant  le  nombre 
d'images  ndcessaire.  Les  diagrammes  suivants 
illustrent  la  mdthode. 


1 

2 

3 

1 

2 

3 

4 

5 

6 

^m 


A    TRANSFOmrATION 


OF 


HANSEN'S   LUNAR   THEORY 


COMI'ARKn   -VSITIl   TIIK 


THEORY  OF  DELAUNAY. 


SOrON   NEWCOMH, 

SUrKUI.NTKNDKNT  AMKKICA.N   i:i'lli:.\|];|{l.S, 


AIDKK     HY 


JOHN    MEIER, 

ASSISTANT  AMKlilCAX  El'lIEMKIUS. 


57 


TIlAXSFOIlMATloV  uF  fl.WSKNS  LINAK  THEORY, 


■a 


I 


TIk-  imni.Ticnl  .•u:ni.utation  ..r'  rlu-  iiuMiunlirios  in  tli,.  uHMm's  , notion  (.x.-ruK-d  bv 
II.VNSKX  was  prolKil.ly  tIk- ^TeaK-st  st.-p  taken  in  ivrmt  tin,,...  tuuanl  pl.-nin^  tin-  tl,.'- 
()iy  ot  tlie  Immr  i.ei-nirhatioi.s  o»  nn  ac-iiiat.-  nnnicrical  l.asis.  It  uas  tlu-  sf.-j,  nlii.-h 
first  reiuU.iv.l  it  r.-rlaiu  liiat  .my  .Hsrivpancv  betuecMi  th.-  tlM-..ntical  an.l  ul.st-nxl 
values  ot  tlic  ni..,,ualiti.-s  pr...hi.-..-.l  hv  rlu-  Min  an.s,-  in.m  s.nnc  utl,,,-  caiiM.  i],m).  trr-.r^ 
111  tli...ry.  'I1,e  tlie-.i-.tical  valii.^.  f,  nhicl,  it  k-tl  nnist  !..•  .•unsidMrd  tlu-  nn.st  a.vii- 
rato  wiiicli  astmnonn-  ij<»\v  jj« .».s.>>t<e>. 

Tlio  only  tlKM.iy  wliich  .-an  f,,niiu-r<-  uitl.  IIanskn's  is  tliat  of  I  )i:i,ai  nav.  H.^ro 
tlie  coodicicuts  aiv  .K-v<.],.j.<,.l  In  ,,,,-{.■>  .-onv.T-in-  s..  sl,,v,lv  that  sonu-  of  tlu-  r.-s.ilts 
are  still  .,  little  .l.Mil.tfnl.  iiotwithstuinlin--  the  -ivat  .-xtent  to  which  the  apj.n.x- 
inuition  was  eamo,!.  It  may  U,  expeet-.l  thai  the  numerieal  theory  on  whi.-h  Sir 
(JKOKcn;  AiKV  is  now  eu-ji-v-l  uill  form  y^  an.,tluT  step  in  advann-,  in'  which  iiotldnjr 
will  Ije  wanting-  lor  tlu-  imijH.>t-s  .,f  accurate  astruuoniy,  so  that  three  tlu-oiies  of  tliT- 
hi-hest  order  (pf  .-, -curacy  will  uhimately  l,e  availal.K-  for  the  eonstniction  of  l„uar 
tables.  The  work  in  .ple^rk.iJ  UAw^r  >rill  uiitinishetl,  the  residts  of  jJAXsh.v  aiul  Dklai;- 
NAY  are  the  oidy  ones  now  avaiiaMr^ 

rnfortnnately.  tlu-  tlie-TV  .,f  FFanskn  cannot  I.e  directly  compared  with  those 
whi(diliave  precc-ded  it.  owin^r  to  fli^-  peculiar  form  oi'  the  variables  in  .vhich  the 
co-ordinates  of  the  m<H.n  an-  .-.xpres.,-d.  In  sayin---  this.  1  do  not  contest  tlu-  proj.o- 
sition  tliat  this  form  h;is  a<ivaiita:r».-.-i.  l!ut,  apart  from  the  cpn-stion  of  its  merits  in 
forin,  it  becomi-s  important  U>  have  the  means  of  making- a  direct  comparison  of  Han- 
sen's theory  with  that  of  hi>  |.re.leees.sors  ami  c(daborers,  who  have  e.\presse<l  the 
co-ordinates  of  the  nu.oii  .iirecfly  in  terms  of  tlu-  tinu-.  This  has  twice  lu-en  partiallv 
done:  by  the  writer  in  the  0.«»/./*.  li^mlii^  for  i86S,  I  (T(.me  LX\"I,  p.  1197,,  ••""h 
indepeiul(-ntly.  by  Schjki.lkkj-i..  itt  a  paper  published  in  1874  ])y  the  Danish  A(  ideniv 
of  Sciem-es.  jioth  dej.eiul  on  data  for  the  transformation  oiven  by  IIansk.n'  1  imself, 
Avluch,  thouizh  tlu-y  may  b,,- ac<-iinite  enou-li  to  -,nve  an  idea  of  the  aj^reement  between 
the^  theories  of  IIanskn  and  1»el4I-xay.  camu.t  be  re^-arded  as  sulliciently  precise  lor  a 
satisfactory  transformed  theory.  The  object  of  the  present  paper  is  to^make  a  ti-an>- 
formation  which  shall  faithfjilly  represent  IIaxsen's  latest  theory,  aiul  be  exju-essed  in 
Jirgumenls  dopending-  directly  on  the  time. 

59 


6o 


TKANSl'ORMATION  OF  HANSENS  LVSWs.  THEORY. 


I'm 


v^    I. 
i:\l'l!i:SSI()N  OK  TIIK  MOON'S  lAJNGITI  UK. 
In  IIax-skn's  tliLM.ry  tlir  iiuu.ii's  louj-itude  is  re|»n.-^-i.j..-.l  in  tliu  followlnj-' form. 

I/,  tli'.»  luooii's  moini  jmniiKily  ; 
I/' .  tilt'  Sim's  lui'iiii   iiiitiiiiily  : 
c),  tlic  (listiuicc  troiii  the  uniU-  to  tlif  ]i('ri;:«'-' : 
(,}'.  tlic  ili>\..i('.(!  from  tile  ikmIc  I(.  the  x'lar  jM-ri'^iee; 
T,  the  lon;4'ituil('  of  tlic  ] n tI ;:■(>(■ : 

r.  tlu;  iH'ceutrii'ity  of  liic  iiio'>iiV  (H-liit.  as  ux-d  l.y  Hax-SKN  ; 
)iS.:,  the  Iliiiiscuiaii  ptM-tiirliatioiis  of  mean  aiK-maly: 
.s'.  tlie  n.insciiiaii  |icrtiirliatioiii>  of  liuitiKle: 
I.  tlie  iiicliuatioii  of  the  moon's  orhit. 


'I'hen  |)Ut,  as  iiuxih'ary  (|uantitu's, 


./ —  elta  (',//  +  i"'^.-),  tlie  true  anomaly 


tan 


.,  1 


I  sin  2  (./ +  o.>}  +  '  tail'      I  ^iu  4    /-^  «>  —  etc 


fhe  retlnctioii  to  the  eeli[)tie; 

,,,  tan  1  eos  (,/'+  m) 

I  —  snr  1  snr  (  /  +  m) 

—  o"  397  ^i"  2  CO 

—  I  ".198  sin  (2//'+  2  co') 

—  o".2S5  sin  (  2  ,'/  —  4  //'  4-  2  &J  —  4  cii'X 

the  iiiei|Ualities  of  this  reihictioii. 

Then,  for  the  moon's  loiigitiido, 

K-/  + ;x+  K-r  I!- 
The  latitude,  fi,  is  <;iven  by  the  equation 


1 


1 


sin  /:/nsin  I  sin  (/+ w)  4- -*. 

In  presenting'  Hansen's  results  in  the  form  <if  a.  complete  and  exact  numerical 
theory,  several  precautions  have  to  lie  taken,  hi  the  first  plate,  all  the  results  must, 
so  fur  as  possible,  depend  upon  or  be  reduced  to  one  and  the  same  hcniogeiieous  set 


i 


TK.VNSI-ORMATION  OF  HANSEN'S  LUNAR  THEORY 


6i 


of  elcnionts.  Tn  tlir  next  pliuv,  tlu.s,.  iiM..,unIiti..s  wl.id,  ..xpn-ss  tl„,  sulutiun  ui  tin- 
prohlci.i  ot  tlii-ce  iMMlios,  nnisi.lcre.l  as  iniitcrial  points,  must  1...  s.^arated  fn.ni  iin- 
quahtu-s  arisinj.'  tVun.  utlicr  sources,  sucli,  tor  instance,  as  tlie  .listan.'c  l,..tn-eMi  ti.e 
moon's  centres  offrmvity  an<l  fio-ure.  and  tiie  eilipticity  cf  tli<.  eartli. 

Tliree  values  of  tin;  eccentricity  appear  in  IIanskn's  tlie(.rv  and  tables: 

(1)  A  provisional  or  ideal  <-ccentricity,  with  which   fhe  ine(Hialities   w-re  ori-n- 
nally  computed.  " 

(2)  An  apparent  eccentricity,  uliicli  he  found  to  represent  the  ohserv.-.l  motion  of 
the  moon's  centre  of  (iuure,  and  used  in  his  tables. 

(3)  A  theoretical  eccentricity  of  the  true  orbit  described  by  the  moon's  c(  ntre  of 
f^ravity. 

Tiiese  three  values  of  the  element  are: — 


(i)     r  —  .05490079 

(2)  r  =  . 05490807 

(3)  '==.05489959 

_  Accordinjr  to  IIanskn's  view  it  is  the  third  value  which  should  be  used  in  c,nn- 
puthifT  the  moon's  perturbations;  ])ut  as  he  actually  used  the  tirst  value,  it  is  tia-  on- 
which  we  should  employ  in  tlie  transformation. 

In  the  case  of  the  inclination  there  are  three  correspondino'  values,  with  an  addi- 
tional complication  arisin..-  from  the  (pu'stion  whether  we  shall  add  to  the  inclination 
Ji  term  in  the  perturbations,  2'.7o5  sin  (//+  a,),  havinj.-  the  mean  arf^unient  of  laTitU(h- 
<is  its  argument. 

Omitting  this  term,  the  values  of  the  inclination  will  be  :— 

(0     1  =  5^   8'  48" 

(2)  1-5'  8'  43".66 

(3)  I  =  5"  8'  39".96 

Here,  again,  it  is  only  the  hrst  value  with  which  we  are  con<  ■  ;    od  iii  the  tran>fonna- 
tion,  because  it  is  the  one  employed  by  IIanshn  in  coin[)uting    '.;  perturl)ations. 

The  IlanseiiiiUi  perturbation  >t6~  is  ai  xplicit  function  of  7,  //',  a)  and  co'.  So 
lar  as  the  longitutle  is  concerned,  our  present  [)robleni  is  to  express./,  R  and  R',  and 
thence  L,  as  explicit  ♦unctions  of  the  above  four  quantities.     If  we  put: — 

2  =.  f/  -}-  uSz 

Ci,  f.,,  ?:„  etc.,  the  coefficients  of  sin  „-,  sin  2^-,  etc.  in  the  development 

of  elta  {e,  z),  we  shall  have, 


f=.^-{-Ci  sin  ^  -f  (',,  sin  2^  +  etc. 


62 


TRANSFORMATION  OF  HANSEN'S  LUNAR  THEORY. 


If,  thon,  WO  put  //  +  mU  tor  ^,  (lovclop  in  |M)\vei-s  (»f  «r5,r,  call  {'\  (f),,  tlu*  part  of  /'iii- 
(lopcndnit  of  i/<'>:\  and  (c,  ,7)^  tlio  (•(.(■niciciit  id'  {u<'^::)'  in./;  wo  sliiiU  have, 

(«',  r/)„  rr  //  +  '\  sin  //  +  r,  sin  2//  -f-  r,  sin  3//  +  r,  sin  4//  +  etc. 

(/>,//),  =  1  +  r,  cos  //  +  2(:,  cos  2//  +  .V',  cos  3//  +  etc. 

1        .  2"        .  3-' 

(c,  y)..  =  —  ,,  ''i  f*ni  //  —  ^  r,  sin  2//  —  ^  r,  sni  3//  —  etc. 

I  2'  3' 

(f,  //);,  =  -  ^  ,  ( ,  cos  //  -    -   (■,  cos  2//  -  ^     T;,  cos  3//  -  ftc. 

-"  »>  *3  *•  *  J 

(f,//),  n-—'',  sin  //  +  etc. 

etc.  etc. 

The  coeilicicnts  r,,  r,,,  etc  ,  lU'c  (Impendent  o)i  tli(!  eccentricity.     Tliu  well-known  ana- 
lytical \aliies.  and  tlu'  ninnerica!  values  obtained  hy  outtin;-' c  rr  .05490070,  are: 

^.,  -  2  c       -  '  c''    +  ^^  r''    =  .10976024  =  2  2639",676 
''•=  =  4  "'     ~  24  '■'  +  {h  ''"  -  -^^-^^ -'^''-^^^  =  7 7^^"- 269 


64 


I  2 


10 


.>  ,/l 


96  480 

'097^5 

960 


—  .00017893  iz:     36". 907 
=:  .00000972  =:        2". 005 

—  .0000005  7  =      c".  I  1 S 


\ 


900 


—  .00000004  ■=.      o  .007 


Tlie  value  of  ;m5^  is  taken,  not  from  IIanskn's  tables,  but  from  liis revised  results 
{riven  in  the  Ihirlff/uiiii*.  They  are  found  in  I'art  I,  pp.  409-411,  and  I'art  11,  pj). 
224,  242,  25S,  and  26S,  and,  for  convenience  of  reference,  are  all  collected  in  Table 
1  of  the  present  [)aper.  In  this  talde  are  •i'iven  alsd  the  powers  ot'  lu').:,  the  coinputa- 
tions  of  which  were  ail  made  in  duplicate,  that  of  the  scjuare  bein;^'  executed  by  t 
inde))endent  comj)uter.> 


wo 


We  thus  have  all  the  data  for  the  numerical  value  of  f,  the  formula  for  which  \> 


Consider  next  the  tirst  term  of  K,  which  we  niay  call  Ki.     We  have 


(0 


K,  =  —  tan-      1  sin  (2/+  2  w), 


which  is  also  to  bo  developed  in  ])owors  of  Hf5,r. 

^  I'lMlcr  lliin  Ullf  n  I'lTciiii'  is  iniiilc  \i>  Hiiiiscn'n  Iwo  papers,  Itiirleijunijihr  lliioirlixcheu  liirichnutiij  tUr  in  dm  Miindta- 
filn  toKji'iviiiKlUK  Sliintiiijni,  in  the  Ablmiidlunijni  dir  kiiniijlirh-mchsischen  OeuUmUuft  dcr  ll'inKiiixdiafkii.     Hand  /A",  Xl. 


TUANSI'OR.MATION  OF  HAN.SI:NS  LI'NAR  TIIKORY. 


63 


Tf  WO  .sul>Htitut.  lur/  ifs  v..,l,u.  in  h-nns  of  .  ..hI  .,  .,,.1  dovel..,,  in   ....vc-rs  uC  . 


If,  =  — tiin-     I  X 


24 


+  .^c 


:\ 


12 


■sin  (—2.;  ■\-2(.i) 
sin  (-    ^-  _|.  2  o) 


+ 


0- 


(^=-s'0 


'c-'+ „r''   I  sill  2  0) 


4 
■^'  +16'' 


j  sin  (       c  +  2  «) 
j  sin  (      2Z  -\-2c0) 


+ 


+  (^4  '■"-  24  '■  J'^'"(     4-4-2^0 

+  J^^^'      sin(     5--  +  -''^) 

+  ,6  '^        •'^i"  (     6c +  2  a)) 

Tf,  in  this  o(jniition,  we  snl.stitute  for  r  and  \  tli.-ir  numorical  valnos  and  tl.on 
ddlorentiate  witl,  respoct  to  ,:,  so  as  to  ol.tain   tl..  .•o.-frn^icnts  of  tl.o  i^owers  of  h6: 


luttmy 


wc  have 


1m  =  1{,  „  +  17,  ,  ^/A',:  -f  Ifi ,  (;/,"),--)-'  -f  otc, 


'm.o  =  —  o".oo6  sin  {—  fj  -\-  2Go) 

—  o".942  sin  (  2  &») 
+  45".62  7sin(     // +  2  oj) 

—  41 1".626  sin  (  2 /y-|- 2  &)) 

—  45"-2''^i  sin  (  3//  +  2  ta) 

—  4". 040  sin  (  4  ^y  -)-  2  Qj) 

—  o".33S  sin  (  5 // -f  20)) 

—  o".02  7  sin  (  6  /y  +  2  &>) 

K,  1  =  +  000  221,2  cos  (  /y-f2  0j) 

—  .003  991,2  cos  (  2  //  -f-  2  f.)) 

—  .000  658,6  cos  (  3  //  +  2  &j) 

—  .000  o;8,3  cos  (  4//  +  2  w) 

—  .000  008,2  cos  (  5  ,*/  +  2  G)) 

—  .000  000,  S  cos  (  6//  +  2  &)) 


•  'TiiWfsof  (his  and  tlio  otlior  ilcvcldiiiiiciils  in  lli..  (■Ili],ti.'  niiilioii  li,i,.  ,m.-«  ;riv.-n  liy  t'rof.-.s<.r  Cayi.ky  in  flio 
M,:m,ni:'<  of  Ihe  Ittnjal  Anlroiiumical  XooWi/.  \,.l.  XXIX,  l.nt  il„.  „buvc  dLM'l..imiL>nt  was  .•xe.ntcl  in.I.'|.rn.l.-ntly  hefort) 
till'  iiitplicability  of  I'lolVusor  Cavi.kv's  loriMiilir  wa.s  ivniarkcil. 


64 


\v< 


TRANSFORMATION  OF  HANSEN'S  Ll-'NAR  TIIKOUY. 

U, ...  =  —  .0001  I  Kill  (  //  4-  2  co) 
-f-  .00399  sill  (  2  //  4-  2  f.i) 
+  .00099  m\  (  3  //  4-  2  a)) 
4- .000 1 6  sin  ( 4  ^  +  2  cj) 

li, :,  =  4-  .002  7    COS  (  2  //  4-  2  &') 

-f-  .0010  COS  (  3  /y  4-  2  <») 
In  the  same  way,  jjuttinj,' 

U..:=itan'i-  I  sin  (4./  4-4'«') 
liuvc  by  siiltstitiiliii';'  Inr,/'  its  viiliu;  in  ,;,  and  (U'VC'l(ii»iny  in  powers  olr, 


i 


sill  (4/4-  4  ^'' 


1 1 


('■   sill  (2  c  4-  4  m) 


Puttinii'  as  betbro, 


—    4r     sin  (3,~4-4<y) 

4-  ( I  —  1 6  e")  sin  (4  ~^  4"  4  <") 

+    4  r    sin  (52  +  400) 

4-  ^-  e"   sin  (6  ,^  4-  4  go) 


11,  -  1:,,,  4-  II,  /M5.:  4-  1{,,,  (;/(5.-y-  4-  otc. 


we  iiml  b\-  snbstitiitino-  the  numerical  vahies  of  I  and  r 

K.J,,  =  4-  o".oo7  sill  {2  //  4-  4  t.:)) 
—  o".092  sin  (3  //  4-  4  &)) 
4-  o".400  sin  (4  //  4-  4  '«^) 
4-  o".092  sin  (5  //  4-  4  r«') 
4- o".oi3  sin  (6// 4- 4  ^) 


—  .000  001 ,3  cos  (  3  ^r/  4-  4  '«') 
4-  .000  007,8  cos  (4  //  4-  4  ^') 
4-  .000  002,2  cos  (5  //  4-  4  Gii) 


The  terms  of  li.o  (»'^c)-  are  less  than  o".ooi. 

The  coeflicient  uf  —  s  tan  1  in  li'  is,  Avitli  suilicient  accnracy, 

cos  (/4-  ai)  [14-  siir  I  sin"  (/4-  «))] 


or 


(  I  4-     sin"  I  J  cos  (  f-\-  00) sin"  I  cos  (3/4-  3  ^)- 

V        4  /  4 


1 

1 

I 


TRXNSI'OKMAiioN  or  HANSKNS  \J\\U    rilKony, 
\W  til.'  (Icv.i.|.i,„.iits,,ft|„.  ,.|li|„i,.  „„„;,,„  ^,,.  1,.,^,.^ 

C.S  (  y-f  ,o)  =  —     '^  r-'  CMS  (-  2  ,•  -f  ai) 


s 


' '"  ( 


•(IS  (  — ,:  -j-  m) 


-{■  ( \  —  r- )  cos  (  ;■  -f-  ^)) 
4-  ('■—  ■\''')  ens  (2,;  +  <o) 

.   9   ., 
+  ^,  '■•  <'os  (3  :  4-  a)) 

+  ^■'  cos  (4:  +  f.>) 


COS  (;,,/■+  ]  a<)  — 


21 


(•-  +  3"') 


—  ;,  '•  (■(  IS  (2-4-  3  <.)) 

+  (1   —  (jr-)  CIS  f  ;  .     .    ;,  M) 

4- 3(' CIS  (4:4-  •  c) 

If  WO  repi.'S(..iit  liy  S  tli(!  (•..(■Ilici.-iii  nf  v  ii,  IJ  ,  that  is, 

S  —  —  tan  1  cos  (./'4-  ^')  ;  1  4-  sin-  1  sin-  (./■4-  a,))  J, 

S  =  S„4-S,  ^,r5,:4-S,(/.r5,;)-, 


and  suppose 
we  shall  liave, 


65 


S„  —  4-  .0000  :;4  cos  (—  f/  4-  r.)) 

4"  -^0495 5  ^'"^  ^' 

—  .0899 78  cos  (  //  4-  U)) 

—  .004936  COS  (2//  4-  (O) 

—  .000306  COS  (3  //  4-  &)) 

—  .000020  COS  (4//  4-  G)) 

—  .000030  COS  (2  //  4-  3  G)) 

4- .000 1  ;6  COS  (3^-1- 3  t^) 
4-  -000030  COS  (4  (/  4-  3  ft)) 
4-  .000004  *■'"*  ( 5  //  +  3  &') 
S,  =  4"  -0900  sill  (   //  4-  (,}) 
4-  .0099  sill  (2 //  4-  &>) 
4"  .0009  sill  (37  4-  co) 

S,  —  4-  .045  cos  (    //  4-  00) 

4-  .010  cos   {  2  //  4-  f«') 

Mjiltiplying-  these  several  exiiressions  by  I1an.skn'.s  s,  we  find  the  value  of  .s' 8,„  etc., 
given  in  Table  11. 
2 


#^:i-'*v.- 


wV**^-**': 


55  TRANSFORMATION  Ol'  llANSICN'S  LINAR  TIir.OKV. 

Collecting  all  tlio  cooHirionts  of  the  i...\vers  ..f  ^mV;,  wc  lind  the  following  expres- 
sions for  the  moon's  true  eeliptic  lon-itii.le,  as  a  function  of /m^  ;  :— 

Terms  Unliti mlrni  n/nS:. 

-}-  22639". 676  sin     // 

-f  776". 269  sin  21/ 

4.  ;/j".907  sill  3// 

4-  2".oo5  sill  4// 

-f  o".i  18  sin  5// 

+  o".oo7  sill  ()(/ 

—  o".oo6  sin  (— //  +  2  co) 

(—  o".C)J,2)     . 

:  •  sill  2  f.) 

(-  o".397^ 

-f-  45". 62  7  sill  (    //+  2  f«0 

—  41  I  ".626  sill  (  2  v -|-  2  c.)) 

—  45".2Si  sill  (3,'/+  2  ^)) 

—  4".o1o  sill  (4//  +  2  f«>) 
_  o".33.S  sin  (5//+  2  a>) 

—  o".o27  sin  (6//+  2  (.)) 

+  o".oo7  sin  (  2//  +  4  r*)) 

—  o".092  sill  (  3  //  +  4  (->) 
-f-  o".400  sill  (4// +  4  ^)) 
+  o".092  sin  (5//  +  4  <-«') 
4-  o".oi3  sin  (6// +  4  &)) 

—  l".iQS  sill  ( 2//'+  2  (o') 

—  o".285  sin  t^// —  4,'/ H" 


f«) 


4w) 


('(icffic'u lit  (if  II '5,:. 

(The  ic. 111111:1  pilillls  llIT  ~i\  llhlrrs  iirdrciiiiiils.  I 

-)-  .109700,2   COS       // 

4-  .007526.9  COS    2  11 

4"  .000536,8  cos  ^n 

4-  .000038,9  cos  4// 

-■f-  .000002,8  cos  5// 

4-  .000221,2  cos  (    //  4-  -  ^•') 

—  .003991,2   cos   (2//  4-    2  f«>) 

—  .000658,6  c(»s  (3//  4"  -  f'^) 

—  .000078,3   cos   (4//  4-    2  M) 

~  .000008,2  COS  (5//  4-  2  r,)) 

—  .000000,8  COS  (6//  -[-  2  <i)) 


TRAN'SroRMATION  OF  HANSEN'S  LUNAR  TlllCORV.  6? 

—  .oooooi,;,  CDS  (3// -f  4  (.)) 

+      .000007,8   (MIS    (47   -f   4  G)) 

+    .000002,2  ('(ts  (5^  +  4r,)) 

+  -  .^, 
I--..  =:  —    .054SS  sill      f/ 

—  .0075;,  sill  2// 

—  .oooSq  sill  3  f/ 

—  .ooooS  sill  47 

—  .0001  I  sii.  (    //  -\-  2  &)) 

+      ■00390  sill    (2//  -f    2  («)) 

+    .00099  sill  (3//  -\-  2  (<■>) 
+    .00016  sin  (47  -(-  2  Mj 

I..;,   —    —       .01  8,^    COS         // 

—  .0050  COS    2  /■/ 

—  .0008   (.'OS   3  // 

+      .0027   l'(»S    (2//  -f    -  &>) 
+      .0010  COS  (3//  -(-   2  &)) 

'II1C  several  parts  of  this  expio-^ioii  tor  L  are  i^iveu  in  Table  II,  oniittiiin-  rlie  fo]- 
lowiii;;-  terms,  wliicli  are,  however,  all  inchuled  in  the  coliinin  i^'iviiiL"-  the  concluded 
coellicieiits  in  L  : — 

1.  The  terms  ot'  L,,,  explicitly  liiveii  in  the  tirst  ol' the  precedinn-  c(jnations. 

2.  The  exiiressions  for  //'V;,  { 11 ''):)- X  "^  ^„  (//'')  .:)-'X  'm,:;-  fi"d  ii/'')-)  X  \<:.u 

The  values  ot'  die  last  three  expressions  are  as  t'ollows.  the  nuniliers  within  the 
parentheses  heiii;;'  coetlicients  of//,  7,  m,  and  cv/,  respecti\(d\- : — 


»  '5  „-  X  IL I 


{,iS,:Yx^^2 


{»'y:fX  1^,: 


—  .001 

sill 

(3, 

1 

01 

0 

0  \ 

—  .002 

sin  (0, 

^ 

0 

—  2) 

—  .00 1 

sin 

(2, 

I 

"1 

0) 

+  .001 

(', 

2 

2 

2  ) 

+  ■0'^3 

sin  [2, 

O 

—  1 

2 

-\  \ 

+  .001 

sin 

{2, 

—  I, 

-  1 

0) 

—  -005 

(-^ 

2 

"> 

2  ) 

+  .002 

sin  (3, 

O 

—  1 

-^ 

2  ) 

—  .002 

sin 

( *-', 

o 

0, 

-^) 

—  .020 

(3. 

1 

0 

2  ) 

+  .002 

sill  (  — 

I,  2 

c 

2  ) 

—  .004 

sin 

('. 

n 

0, 

2) 

—  .005 

(4. 

-> 

T 

>  ) 

+  .(304 

sin  (c. 

'> 

-» 

0, 

0  ^ 

—  .002 

sin 

(2, 

n 

0, 

2) 

—  002 

(.- 

I, 

4, 

0) 

—  .002 

sin  (2, 

_  |, 

4, 

-^ ) 

-f  .002 

sin 

(2, 

2 

4, 

-2) 

+  .002 

(4,- 

I, 

4. 

0) 

—  .002 

sin  (3. 

^ 

4. 

—  2 ) 

+  .003 

sin 

(3, 

—  2, 

4, 

--) 

+  .001 

(.i, - 

I, 

4. 

0) 

—  .002 

sin  (4, 

-^ 

'>, 

—  6) 

+  -003 

sin 

<4, 

•^ 

4, 

-:^) 

—  .003 

(4. - 

'1 

6, 

-  2) 

-•  .002 

sin  (,■', 

-<j, 

C), 

—  6) 

•j-  .002 

sin 

(5, 

0 

4i 

--^) 

+  .016 

sin 

(5, - 

T 

6. 

-2) 

—  .002 

sin  (2, 

-6, 

4. 

—  6) 

—  .001 

sin 

(1. 

—  6, 

4. 

-6) 

+  •013 

sin 

(0,  - 

0 

6, 

-  2) 

—  .002 

>i"i  (3. 

-  6, 

4, 

—  6) 

—  .001 

sin 

(2, 

—  6, 

4. 

-6) 

+  .002 

sin 

()•,- 

T 

6, 

-2) 

—  .001 

—  .001 

sin 
sin 

(6. 
(7, 

—  6, 

—  (^, 

-6) 
-6) 

68 


TRANSFORMATION   OF  IIANSKNS  LUNAR  THEORY. 


Til  T.-ililc  II  tlic  (•(tliiiiiu  "Sum'"  contains  tlic  sums  of  tlio  torms  actuallv  iriven  in 
the  pivccdinn'  coluunis  of  the  talilc 

TIk'  uc\t  column  ^ivcs  the  com]ilct('  cocllicicnt  of  cacli  term  in  the  ecliptic  lon^i- 
tmle.  and  is  t'ormed  l»y  addiiiu' to  the  column  "Smu"  the  omitted  ti-i'ms  just  referred  1o. 

The  last  c(dunui  j^ives.  for  the  larger  terms,  the  elements  which  the\'  principally 
contain  as  factors.  If  these;  elements  he  chaniied,  the  coellicients  nuist  he  chiiniied  by 
corresponding;'  (plan titles. 


vS  2. 


IJKDrCTIOX  OF  THF,    IMM'.CIvDrNd    KXIM.'KSSIOXS   TO    ('XFFOini    KLK.MKNTS,  AND 

COMI'AHISON   WITH    I>KI.ArXAV. 

The  coeHicients  of  the  precediui>-  incfpndities  contain  as  factors  certain  elements 
for  which  different  investigators  adopt  ditl'ereiit  values.  It  is  essential  to  a  (dear  pre- 
sentation of  results  that  they  should  l)e  reduceil  to  a  uniform  and  W(dl-define(l  set  of 
elements  havinii- j^iveii  v;diH-s  We  tlierefoi-<'  coinmeiu'e  hv  i-educlnu'  tin;  theorie.s  of 
both  Han.skx  and  Dki.ai  NAY  to  such  a   system.      The  (dements   principally  referred   to 


are 


mam 


(a)  The  ratio  of  the  mean  motions  of  the  sini  and  moon, 

(fj)  The  lunar  eccentricit\'. 

{)')  'Hie  solar  paralla.K. 

('M  The  solar  eccentricit\-. 

(f)    The  iiudination  of  the  moon's  orbit. 

Iveally,  all  tlu'se  (dements  are  coiitaiiieil  in  all  the  ineepnilities  in  a  very  complex 
ler.  Ihit  there  is  so  little  doul)t  about  their  true  numerical  vahu's  that  it  is  only 
necessary  to  take  account  of  their  (dian.u'es  uheii  they  appear  as  factors  in  coefHcients 
of  coiisideral)le  man'uitnde.  The  extent  to  whiidi  ea(di  tei'm  is  affected  can  l)e  nai^'hly 
seen  from  its  analytic  e\pressi(m  -iveii  by  1  )ki..-.i:nav  at  the  end  of  hi.s  n,orir  du 
Moitniurnl  <lr  la  Liit/r,  'i'onie  II.      We  take  up  the  several  elements  in  oi'dei'. 

{a)  Untiu  „f  iiH'ui/  iiHiliniis.  This  (dement  is  so  certain  that  no  reduction  need  be 
made  on  ;U'counr  of  it.  It  is  true  that  theoretical  mof'ioiis  of  the  lunar  n(!(le  and 
peri<4('e  imist  implicitly  enter  in  coniKction  with  this  (dement.  Mut.  fn.m  a  row^h 
examination  (,f  I  Iansf.n's  integration   coeliicicnts  on   pp.  350-352   of  his    /fmirf/iirif/'^  J 

do  n..t  flmik  any  (d"  the  laruer  coetlicieiits  will   be  aOected   by  as  nuudi  as         '         of 
»i    •  •  ,  ,     .  "  100000 

tiien-  ennre  amount  by  any  adunssible  (diaiinc  uf  these  motions. 

(/;')    Errndnvitii  of  mu,„>\  orhif.    The  eccentricities  used  by  the  tw.,   investijrators 

are   not  directly   comparable,  but   may  be  nio.st  convenientlvVompared  l)y  nHhicing 

each  to  the  coetlicient  of  .7  in  the   expressi.m    for  the   moon's'  ecliptic  lonnjtude.      J)i.> 

i.Ar.VAV  uses  Aiuv's  value,  niven  in  his  la.st  paper  (.n  the  (dements  of  the  nmon's  orbit* 

Il.VNsi  N  corrected  his  eccentri.dty  for  u.se  in   his  tables,  as  alrei.dv  mentioned.     Tl 

writer  (^tained  a  sniall  but  w(dl-marked  c.,rr(.ction  to  IIan.skn'.s  yuiue  from  the  Green 

•  .Nli'inoirs  lioviil  AHtioiKiiiiical  Socioty,  Vol.  X.XI.X. 


le 


TRANSIORMATIO.N  OF   ll.WSKXS  LUNAR  TIIF.DRV.  69 

wicli  ol)sorvi:tioiis    1846-74,  niid  tlic  AVasliiii;,Hoii  Dbscrvatioiis   iS62-'74.     Tlio  four 
values  of  tl"0  coi'lliciciit  in  (|ncstioii  arc: — 

AiiiV,  used  hy  Dki.ai  NAY,      .      _      .      .  22639". 06 

Hanskn,  used  ill  'riicor\-,       ....  22637".!  ^5 

I  Ianskx,  used  in  TjiIiIcs,         ....  22640".!  ^ 

CoiTcctcd  \alnc  found  in   1876,*       .       .  226:;o".v8 

Altliouj^-h  tlicrc  is  110  ivasonaMc  doiiht  that  llic  ('('ccntricitx-  ..f  1I\nsi-:n's  tables  i-e(|uires 

a  iieg-ative  coi-i-cction,  it  will  Ix'  adopted  for  the  ]air])oscs  of  coinpai'ison   Ijceause  it  is 

tu)\v  the   standai'(l   ot   the   cplicniei-idcs   with    which    ^niis(M|UciU   coinpai'isons   unist  Ik; 

made.      All   the   tei-ius   ha\  iui;'  c  as  a   eoellieieiit,  nnisi   thci'cfoi'e   hr'   ineivased  Itv  tli(^ 

factor 

.00000728 

zr  .0001  ^20, 
.05490 

and  those  liaviii;^'  r'-'  l)y  d(Uil)lc  tins  I'actor.      The  coctlicients   in  r  nnist,  in  1)i:i.ai;nav',s 
theory,  bo  increased  hy  the  factor 


I   .09 
12639" 


.000048: 


{y)  Solid-  juira/ld.r.  II a\si;n\s  theory  does  not  set  out  with  a  defiiute  solar  iiarallax, 
but  with  a  ratio  of  the  in<'aii  distanci's  of  the  sun  and  ukmiii,  which  ratio  ayain  is  not 
the  usual  one,  because  IIanskn's  h  and  n'  aic  the  same  functions  ot' the  motion  of  mean 
anomaly  that  the  i'sual  '/  and  n'  ar(-(iftlu'  sidei'cal  mntinns.  W'c  must  tl.ei'efoi'e  adopt 
an  indirect  pro:  ess  foi'  iindiiiL;-  thr  i'clatii)ii  n\  solar  pai'allax  and  paralhn  fie  eijuation  on 
his  theory.  He  finds  that  his  thcoi'ctical  coctlicimt  has  to  be  nniltiidied  t)\'  the  t'actoi' 
1.03573  b)  make  it  aLi're(»  with  obseiAation:  and  then,  in  ^^  260  of  his  l)(ir/c(/Hi/(/,  he 
deduces  the  solai' parallax  8 '.(1150.  DividiuL;'  this  paiallax  by  the  pn.'cedinj^' factor, 
we  coindude  that  the  [larallax  ol  his  theoi'\-  is: — 

8". 6085. 

In  turnini^' his  tlieorv  into  numbers   I)ki,\i".\ay  used   8". 75.     The   parallax  to  which 
both  theorii's  will  be  actualK  i-eiliii'cd  is: — 

8". 848. 

Hence,    TIansi-.n'.s  tei'nis   hasiuL!'   the   pai'allax   as    a    factoi-   inust  be  inci'eased   by  tho 

factor 

.02785, 

and  Di'.f.AiNAv'.s  by  the  factor 

.on  20. 

('^)  7'lie  solar  <■( cri/tr/iifi/^    The  solar  eecenti'icitx'  ot'  1Iansi;n\s  tlieorv  is: — 

c' =:  0.01679226  (I'^poch   !8oo). 


J'ii|ii'r.-i  piililislicil  li,\  till'  CiiniiiiiK.siiui  or;  llic  'liuiisil,  (if  \'i'mi,s.     I'uit  III. 


ro 


TRANSFORMATION  OK  IIANSKN'S  LUNAR  Tlll'OKY. 


!)i;LArNAv  uses  la;  \'ki{Kiku'.s  value  :- 


c'  :=  0.0167 7 1 06 


(Kpdcli  1S50). 


Ill  strictness  tlicsc  twn  values  are  not  coniiiarable.  owini;'  to  the  diilerenl  form  of 
Hanskn's  solar  tlieorv:  l)iit  since  IIanskx  ne;^lects  |)erturl)ations  of  the  earth's  motion 
in  his  liniar  theor\-,  it  ina\  he  assnnie(l  that  there  Mill  he  no  (lill'erence  between  the 
form  in  which  the  eccentricity  enters  into  the  two  tlieoi-ies.  If  we  carry  Lio  \'i:iiHii;K"s 
eccentricit\'  hack  to  iSoo  with  his  secular  \ariatioii,  we  shall  ha\e : — 


0.01679228 


(K[)(K'li  iSoo). 


This  iiia\-  he  re^anlod  as  a!»olutely  identical  with  IIanskn'.s  value  for  tin*  same  e{)ocli. 
So,  ailoptiiii;'  I  Soo  as  the  epoch,  we  lia\e  oiilv  to  iiurease  Dki.ainav's  coeilicients  in 
<■'  hv  the  hictor 


.00002122 
"707677^ 


.001  205. 


Or,  we  ma\-  re(lnc(;    Hansen's  values  to    iS^o  In-  dividiiii'-  them   hv  1.00126s,  when 


0) 


lie\'  will  lie  comiiaraole  wi 


th  I) 


KI.Ai'.N.W  S. 


lie  riieories  o 


f  IIanskx  and  hKLAiNAV,  thus  reduced  to  a  uniform  and  consistent 


set  of  elements,  are  <iiven  and  compai-ed  in  Tahle  III.  I )i;lai:n'Ay'.s  results  are  fre- 
(juentlv  doul)tt'iil  h\-  a  small  fraction  of  a  second,  owin;^-  to  the  slow  converi^eiice  of 
the  seri<'S  in  powers  of  m,  and  the  table  has  been  arraiij^ed  so  as  to  show  the  extent  of 


lie  iiiicerramtv  tliiis  ans^ui;'. 

l'"ollowinLi'  the  indices  expressing;'  the  arguments  are   LiiNen.  lirst,  Hanskn'.s  coeiH- 


cients  formed  from  the  \aliies  in  T 


lie  1 1  b\'  iniiltij)lyiiin'  ii\- 


lie  a 


ppidpriate  I'actors  tor 


reduction  alreadv  ui\eii.      Thev  are  onl\-  iiiven  to  o".oi,  but  should  the  thousandth  of 


se 


ccnids  lie  reqiiiivd  they  are  readiK'  obtainaljle. 


'I'll 


correspoiidiiiii'  coeflicieiits  of  Dklai.'NAV  are  (lei-i\ed  principalK'  from  his  | 


ire- 


.sentation  of  niimerical  results  in  the  additions  to  the  ( 


niiUtitssinicc  I 


On 


11  to  21  of  that 


paper  are  j^'iveii   the  si 


/c.s   T('i;iji.s  tor  1869. 


iinis  of  the  terms  in  each  coelHcient 


which  were  actually  computeil  by  him.      The  parallactic  terms,  as^'iven  bv  Delainay, 


are  s 


till  t 


o  lie  multiiilied  b\' 


1  -  ,\ 

i  +  A' 


A  beiiiir  the  ratio  of  the  mass  of  the  moon  to  that  of 


the  earili.      rutlinu',  with  Hansen,  A  —      ,  the  coellicient 

80 

rected  for  this  coeflicient  and  for  diiferenco  of  element 


will  he 


8i' 


Tl 


le  sums,  cor- 


th 


s,   are  f^wvw   m  the  column 

fli 


I>('/((Hi/ni/  (\).  Had  all  the  appreciable  terni.s  been  actually  computed,  tlie.se  coetli 
cieiits  would  have  been  the  delinitive  ones  of  Dei.aunay's  theory,  lint  it  was  fre- 
([iiently  tbiiiid  that  the  terms,  even  of  the  ninth  order,  where  the  development  c(!ased, 
were  still  appreciable:   it  was,  therefore,  necessary  to  estimate  the  ))robable  sum  of  the 


0111 


itted  terms  of  lii"lier  orders  from  the  law  of  tl 


K!  series  as  (deserved  in  the  terms 


actually  computed.     These    estimates    can    have    no  true    niatheinatical    fouudtitiou, 


TUANSI-ORMA  riO.N  OF  IIANSENS  LUNAR  TIIKUKV. 


71 


1)Ocanso  tlicre  is  110  proof  of  tlio  iictiiiil  law  of  the  scries.*  .Still,  tlioro  is  a  liii)'li  doj^Toe 
of  probability  in  favor  of  each  one  iicinn'  ■•'■  '•'"^'^  "  ''"'l*'  ajipi'oximatiou  to  the  truth. 
A  rigorous  coniputation  would  probal)ly  show  that  a  majority  dilfcred  less  than  '  of 
tlieir  amount  from  tlie  true  \alues,  though  here  and  there  one  mij^'ht  be  found  entirely 
illusory.  The  coellicicnts  of  l(in<iitudc,  nioditied  by  these  estimated  additions,  arc 
f>'ivcn  !)}•  1  )Ei,ArNAV  on  paj^cs  38-40  of  the  paper  rct'erred  to,  and  are  reproiluced,  with 
the  necessary  corrections  for  chaiin'cs  of  elements,  in  the  colunm  Dc/aiUKi//  (2). 

The  ditference  of  these  results,  ^iven  in  the  next  colunm,  is  the  correction 
ai)])arently  applied  l»y  Dklacxav  for  tlie  uncomputed  terms.  It  will  bi- noted  that  we 
liave  no  indei)endent  statement  of  these  tei'uis  to  refer  to,  ami  can  onl\'  infer  their 
values  fnun  the  ditfereiu'cs  lietween  the  printed  I'esults  (1  )  and  (2) 

Finally,  we  have  the  difference,  I/ai/^n/  minus  DihuDitiii  (2)  >liowin^-  th(^  dis- 
crepancies still  outstanding;-  between  the  two  theories  l'',ach  one  can  jndi^e  foi'  himself 
liow  far  these  discre[)ancies  arise  fi'om  the  uncertainty  of  Dklainay'.s  senn'-empirical 
corrections,  and  how  far  tVom  errors  in  the  two  theories. 

.  One  or  two  terms  are  woi'thv  of  a  special  examination,  and  amonii' tii(>e  the  par- 
allactic C(piation  takes  the  liist  I'ank,  a>  upon  it  depends  the  value  of  the  solnr  [)ai'allax 
to  l)e  derived  from  a  ^^iven  observed  value  of  this  ecpiation  Arrani;'iiij.;'  1  )i:1jAII.nav\s 
terms  according-  to  the  power  of  ///,  which  enters  as  a  factor,  the  result  will  be  that 
<4'iveii  b(dow  under  the  head  I',.  Dklau.nav  omits  terms  in  y-  after  ///',  ami  t(.'rms  in  r 
after  in''.  Corri'ctinii'  the  result  i'or  an  estimated  valm-  of  tliesi'  terms,  derived  b\'  in- 
duction, we  shall  have  those  i;i\-en  umler  the  heail  !'_..  It  will  be  seen  that  the  terms 
follow  a  nearly  reyular  law  up  to  in'\  but  that  hi'  deviates  from  this  law.  Assumin<>' 
this  term  to  be  in  error,  and  estimating'  the  valr     ot'  it  and  the  higher  terms  as  those  of 

a,  ycomcitrical  proi^-ression  with  the  ratio         we  have  tln'  residts  1';. 


Po 


p. 


Terms  In  //^     —     73". 1760      —     73". 18     —    73". iS 


iir 


Sum 


\   -o^J-i   —  34  -o 


o 
I  2  .01 


34  -30 
12  .01 


III'    —  12  .0082 

w/'  —   4  .()8i2   —   4  .50  —  4  .50 

///"'  —   I  .9815   —   1  .89  —  I  .89 

111^'    —   o  .7122   —   o  .72  —  O  ."JZ 

—  o  .48 


III'    —      o  .381  I 


o  .72 
o  .3S 


127  .242- 


I  26".c)S     —  I  2  7".oS 


Our  choice  nnist  lie  between  the  results  P.  and  1';,.      ff  we  ado])t  the  fornuir  we  may 
add  o".26  as  an  estimate  of  onnttiu"-  terms  <,^iving'  : — 


1' 


r=-l27".24;   P'=:J^P=:-i^4". 


TO. 


•  W  tiKiy  lie  iviMiirli.'d  I1i;il  in  llir  sni.'s  U>v  \\\i-  .-niiliir  ;iici'I,t;iI  ion  Ori.u  \\1    I'l I  (Iw  Iniiis  ol' ii  lii^liii- (Udcf 

ac'liiiill.v  111  cliMMHi'  llii'ii-  -^iK"!  (liivt  ll.v  ciiiiliiir.v  m  ll ^liiii.ili'  xvliiili    wmilil  li;ivr   luiii   tMiiiir,!  rmiii  \\\<>s,-  of  ;i  liiwrr 

(iidiT, 


TRANSFORMATION  OV  HANSEN'S  I.INAR  TIIKORY. 


If  we  iulopi  tlic  latter  we  li;iv(! 


P  rz  -  i2;".o8  ;         1"  =  :,;   1'  zz  -  I  23'. 94. 


Si 


Miiltiiilyiii;^- l»y  the  eoellicieiit  i.oiu  to  reduce  to  the  i)ariilliix  8".S4S  the  result  will 
be:—   ' 

(-^)         -i-^5"49 
(3)         -  i^5"-33- 

IIankkn's  coetHcieut,  —  i25".43,  I'mIIs  hi-tween  these;  results  and  may  he  regarded  as 
certainly  corn.'ct  within  less  than  o".i. 

The  other  term  referred  to  is  that  ilependin<i'  on  the  ar;^-ument: — 

of  which  the  principal  parts  of  the  coellicienr  are,  in  Dklai'NAv's  th(M)ry, — 


3(1  order, 
4th  order, 
5th  order, 
6tli  order, 
7th  order, 
8th  order, 
9th  order, 


Do 


.10 

-     5"-^o 
+  10'.  1 5 

+    9"-34 

+    5"-^2 

+       2".90 

+     i"43 


Dhi.ai'nay  seems  to  have   taken  i".i8  as  the  ])r<ibal)le  sum   of  the   omitted   terms, 
whereas  they  should  have  been  taken  as  o".94  to  agree  with  Hanskn. 


S^  3- 
LATITLTDK. 


Taking  Hansen's  expression  lor  the  moon's  latitude  :- 

sin  /y  =  sin  I  sin  (/  +  gj)  -f  s  ; 


the  first  step  is  to  form  the  expression  sin  i  f  -{-  m)  in  terms  of  //,  ro,  etc.     This  may  be 
done  in  two  ways.     By  the  first  we  express  the  required  (piantity  as  a  function  of  ^,  and 


TK  ANMOkMATION   <  >K   IIANSKNS  1.1   N.\K    TllloKV. 
tlK'H    put    <l     I-  /mS;   r,„-  ,/    ,.,,„1    .!..v,.|,,p   j„    p.nvcis    ,,('  i,<S:.       I'.v     tllr    lll..,„y 


lit'  cllijitic 


sill   (  /'-I-  r.. 


Oj; 


—  ,    r    sill  I  —   >  .;  -[■  Ml 

9210  ■        .^  •     1         / 


>lli    (-  4,;    +   M) 


-^(~,:s''--k'o"';)-'^---^-) 

-(  -  ,,'^  +  l^  '■'')  ^i"  (-  2.: -fa;) 


r  sill  r.) 


+  (3X4'        19^ '■")'^'"  ^5-  +  -) 


Si 
40 


r'  .sin  {6 .:  +  (>'') 


+      ,-    ,,      '-  sill     7,r  +  m). 

« 

Tl"  we    now   siil)-.tltiiii_'  fVtr  .-.  »/ -|- >"*> :.  t">r  r  its  niiiiu'rical    \;ilni',    and 
[mltiiii;' : — 


i|i-\  fluli. 


we  sliall  liave 


^in  I  .-in  11'+  (r   =  V  --p  F,  ,><^  :  +  F_,  ( i,  <y:)-  +  F,  (n  'y:)'' 


F„  ( ill  arc    =  —  o'  o  I  2  sin  ( —  3  //  +  .'.>) 

—  o  -255  sin  (—  2  //  -(-  r>)) 

—  6'  .96S  sill  ( —     (/  -f  m) 

—  ioi5".S34  sill  f.) 

-f  eS447".342  .sin  (         // +  r>)) 

-f-  roi2".oii  sin  (      2 // -j- 6,j) 

+  62  .45S  sill  (      3//  +  r.i) 

-f-  4  .061  sill  (     4 /y  +  <i>) 


74 


TKANSI'ORMATION  t)!-'  HANSEN'S  M/N  \R  THEORV 


(ill  Jirc)  (CoiitM)  —  + 
+ 
+ 


o".2J2  sit)  ( 
o".oig  sill  ( 
o'  .001  ^•ill  ( 


F,,  (ill  radiusi  —  —  .(.")0(>  oooi  sin  (—  i  a -r  ot) 

—  .000  00 1  2  sin  ( —  2  ,7  -f  «•>> 

—  .000  o^^3S  sill  (—     'J-^*^) 

—  .004  9249  >iii  ''' 
+  .o.S9^352  sill  ( 
-|-  .004  90<j4  sill  ( 
-|-  .000  ;i02S  >iii  ( 
-j-  .000  0197  >iii  I 
-f  .000  0013  >iii  ( 


i7  +  «> 

3,  ff  +  'y) 

4  <•/  -f  <y> 


A>  a  dice 
adoptcMl, 

Then 


-}-. 000  OOOI  sill  (6//-]-") 
F,  —  +  .000  0002  COS  (—  3  fi  -f  <w) 

4-  .000  0025  cos  (—  2  7  -f  **'> 
+  .000  03  3  S  CI  tS  [—       ff-T  f^} 

+  .089  4352  cos(  .7  +  *^) 

+  .009  S I  27  COS  ;  2  '/  +  <") 

+  .000  90S4CI.S1  3  5r4-i»> 

4-  .000  07 S8  cos  (  4  .*/  -r  a>> 

4-  .000  0066  (MIS  (  5  .'/  -f  ^) 

4-  .000  0005  cos  (  6  ft  -r  <") 

!•'._,  —  4-  .(H/O  02  sin  ( —     //  -r  &-'» 

—  .044  yz  sin  (         '/-+-  '*•''' 

—  .009  S I  sill  (      2  //  -4-  <w> 

—  .001  3O  .-in  (      3  V  4- oxj 

—  .000  16  sin  (      4 '/  —  «^* 

—  .000  02  sin  (       5  </  ~  oi) 

F;,  iz:  —  .0149  cos  (      7  4-  <y) 

—  .0065  cos  (  2  ,'/  4-  <^'* 

—  .0014  cos  (  3  ,7  4-  OJJ 

k  upon  the  value  of  sin  i  sin  (./'4-  wj  a  >ec<.]i(i  uieth*>fl  ot"  coniinitin;^-  it  was 
as  toHous.      li"t  lis  put : — 

<yi'—/—'i. 

sin  (,/'4-  f'A  ■=  sin  f  ,7  4-  ro  4-  6/} 

—  cos  'S  /'sin  (v  +  '*ij 
-f  sill  ("i./cos  {</  4-  <^'j- 


TRANsroKM  Aiiox  ,,|-   ||  sssKNs  I.l'NAR  Tllli.KY 


75 


Fn.M.    tl...   ,,..,„..,•!,..]    vain.,  of  <\r  .Wr.Ay    .iv,.,,    tl.-   p.uMS   uf  tliis   .,,nntitv   w.r. 
tonued,  aiul  tluMicc  its  cusun-  and  sine  tn.n,  the  luimnla.  :— 

cos  'W  =     I     —       ■       4-  (>ti. 
1.2 

sin   <S  /  ~  ,S  /  ~     ■'     J..  ,,!,._ 

1.2.3 

'J'licsp  ..xpn-ssions  wciv  fln-n  nnilti|.li.Ml  l,y  t\w  sine  and  cosinr  ..|'  (,/  +  m). 

'I"l'<'  incan  dilV.ivncc  Ix'tuccn  the  cM-iHcicnts  in  sin  I  sin  (,/  + r..)  Innnd  l.v  fli.. 
two  nictlK.ds  was  l(.s>  than  (i.e.,;,.  the  lar^vst  one  hitin^'  ".oiu. 

Addiii-'  IIanskn's  s  In  this  cxpicssi,.!!  we  havi-  the  vahu-  <.!'  sin  //.  'I'hfu  A  itself 
is  (ihtaincil  l.y  the  inrniiila 

A  zzrsiii  /y-f  '  sin'  /5' +   ■''  sin"  /i. 

The  pniicipal  |.arts,,|' /;  aiv  o-ivcn  In 'I'ahl,-  |\\.,f  which  the  rulunnis  .L-t'emnj^- to 
IIanskn's  tliciirx   sccni  lu  nccil  no  cKijIauatiun. 


vs  4. 

iMJtrcTioN  (»i-Tiii;  i.ATin  Mr;  ani>  comcaimson  with  kiilai  nay. 

.Ml  the  terms  i\\'  the  latitude  CMiitain  the  inelinatinn  ol'  the  niDon's  oi'hit  as  a  factor, 
and  are  therefore  to  he  niulti|ilied  liy  such  a  constant  coellicient  that  tin'  |)i-inci|)al  term 
ot  the  latitnde  shall  aLiree  with  ol)>er\  atioii.  'The  transt'oi'ined  expressions  of  IIanskx, 
L:i\-en  in  Talile  i\',  leail  to  a  consistent  theory  in  which  the  otdficient  of  the  |ii'iiicipal 
tenuof  the  latitmle  is  iS4oV'.24.S.  The  expressions  ,,f  Dki.ai  xav  also  lead  to  a  theor\-, 
in  which  tliis  coellicient  is  iSpx'.jo.  i'lacli  of  these  is  to  i)e  multiplied  li\-  sncii  a 
iactoi- as  shall  reihice  it  to  the  \alue  iniplicitl\-  adopted  in  I  Ianskn's  tables.  There 
JIansk.n  adopts  : — 

1  =  5"  S'  39". q6, 

which  is  less  hy  S'.oi  than  that  of  the  theorw      Hence,  from  this   alone  wonld  tVdIow 
the  correction  : — 

—  S".o4  sin  ( ,/'+  '«')• 

Hut,  the  tables  contain,  anioiiL;'  the  pei'Inrliatioiis,  two  terms  which  depend  maiidy  on 
the  same  ariiinneiit,  namely  • — 

2". 705  sin  (./"J-  a>), 
wliicli,  developed  by  putting;' // +  2  c  sin  //  for./i  appears  as  a  perturbation,  and 

3'''.7o  sin  (//  +  f.)). 


„^^  linNSFOiniM-InN  nl    IINNSKNS  I.i;N.\K    lliroKV. 

l.li..lM..nnl.u,.l,o,l,.. ..i.u.ni n,Hs.,r(l.u,v.u,l..r,n.i,v..r,l,. 

„„„„,      Tl..  >Hin  of  llu.  Iir>t  two  ..xpivsMnn.  l-m.-  ^Irvlu,,,.!.  l..rnHK. 

_  5".;,  ID  >ill    *,'/  +  ''') 
-fo".29.:;  sill    M 

—  o".:g2  sill  (J '/  +  f«'.). 
Ad.liii- tlir  lliinl,  111''  I'nu  ill//-}-'.'  will  l'."'"iiic 

-     l".6l(;  sill  I//  +  '.'^. 

W,.  .iv   iw.   ron..,.nH.,l    .ill:   iIh'   -n.-   in  -  .H'l  ^ '/  f '.-■      'Hi.  ;:iv;il..r  |,;.rt  -f  ih^ir 
,,,,,,,,,,,,,,,,,,,.  ..,.n>i,l,.,va  as  ;i  r'-  l-'-tnH-.-H.  .ln.M,Ml,Mi..iiv  ul    U^ 

in.plirillv  .•..il.i.H.l  ill  111.  Ml,l..  Inil  iiu,  h^lunuin,'  U,  ih.  pn.M.m  ut  tliivr  InMirs. 
Willi  the  liist  cinvctiuii  tlu'  t.-nn  in  '/   |-  '•'  Immm.uics 

iS.jOi  ■.'):o  sill  (.7  +  '«'). 
.vliicli  i-  tli.-.'H.'llinMil  iiiipiinlK cMiPiiiMMl  ill  ll\N>r.N'sl;il.lrs.  ,,,.,., 

T„,lii>  ,li,.  writ.rtniMHl  :.  mnvrti..,,  .4-.  ,'.15  iVuni  (  Mv-iiwi.-l,  ;ni.l  ^\  i.slini^'ton 
..h.Tvntioiis  1SM.-7.,.  iMit  it  will  1„.  ,v,;,i,n.i  witliuut  riiniiuv.  II.mht  ;i11  llir  rnHh- 
ri..|it>  in  11ansi,n->  /A  fis  -ivn  in  T;ihl.'  IW  mv  to  1h.  iliniinis!,..!  l.\   tlic  tiirtor 

.ooooSS. 

iiud   thoM'  nf  l>.  l.M-NAV   .IIV  tn   lie  inrrc;l<C(l   li\    tllf   liu'tcr 

.000020. 
•n„.  „.nns  In  ,   .•nnl  -'  ...v  to  !„■  iiiodilicd  liy  tlic  snnic  .■orniciMits   .s  in  ill.-  .-isr  ..ftlic 

l,,„,_,iln.l.'.     TIm Iv  h.iMn>  wliicli  will  I-  iiltrcrlMlilv  uWrvwA  liv  tli.'  cluin-v  ..fr  rm' 

tlioM'  (l(M)fii(lin:.;'  nii  <.>  .■iiid  :  .7  -j-  '■'. 


'I'Im.  1 liii.'atiniis  hen-  iiKlicat.Ml  liav  not  hccu  niadr  in  tin-  ivsnits.  l.c-aiisc  tli-y 

iuv  SI.  >ii-lit,  and  alVi-cl  su  lew  tiTiii^.  tliat  carii  ,iiic  ran  make  iImmii  lor  liiiiiM'lt. 

I'lic  ciluiini  Ihhunniii  (  1  j  (■..mains,  as  l.ctorc.  tli.'  mum  ..I  tli-  l.aiiis  a.tnally  .•<.iii- 
niitfd  \)\  hi-.LAi-.sAV.  am!  -iv.-ii  liy  liiin  in  tin'  Cuiiiniissiuins  ilr-<   I'viiij^s  l.ir  iSog. 

In  Volmiin  Ihhniinni  ;  :  i  In- fli.a.'iit.-  arc  .•..iMvrt;'.!  l.y  the  lii-li.'r  Utiio,  of  wlncli 

the  value  lias  liccii  .•stimal.Ml  l.y  iinlii. •lion,  1>  K  iN.v  liinix-lf  .li.l  ii..t  -i\«-  tlu'Sf 
iul.litiuiis,*!;.)  that  they  liad  to  he  cstiinatcl  '.y  llii'  writer. 


vN  5. 

I'.\|;AI.I,AX. 

IIwskn's  tli.'orv  Li'ivcs  till-  pi'i'liirliatioii^  of  ilic  natural  jo-'ai'lllnn  of  tli.'  111. ton's 
railiiis  v.M'tor.  wlii.'Ji  aiv  tin-  n.'Li'ativ.-  ..f  tli.'  pcrtnvl.atioiis  of  tli.'  l..,^'antliiii  sine 
narallax.  The  valiif  ..f  ir.  in  sccnn.l.-  ..I'  arc,  i>  foiiinl  in  tlic  l)iir/r</ini'i,  I'art  1,  pa^cs 
409-41  1,  ami  i'art  11.  paii'fs  224-226,  25S.  ami  2OS.      'riic  in.it.ii's  parallax  /i  i>  .u-i\cii 

1)\-  IIanskn  uiul.T  til.-  form 

.  I)  (,  .,|-,.  cos/) 

l..n-  sni   y, -!<.-■  ^,  (^j_^,:^  —  '". 


11'!  ANSI  oUM.\l|(i\  oi'  ||\Ns|:.Si's  I.iNAi;    l|||;.»RY, 


n 


'"    "''''■''     "    I-    tll<'    I'iMlillS   nl'    tllC     ..artll     lit     IIm'    lltilinl,.   nf    u|,i,.|,     (llr   >ll„.    is    s,f\.A\xA>l 
','"■    '"■>   Hl.-.ll,    .liMMIlrr    il:     l!„.     1Ln>,,m,,     ll.ru.v.    SvMrl,     i-,    ,|;ii;.,Vlll     n,    .h.|llliliull 

''■ ''"•   ""•""  'liMMiHT  n{  tlir  (,r,|ii,,.,rv  il ri-s.  "  li    is   not.    Ii.,wvcr.  iHTf^-.irs'  t.. 

'''■''""■'■    ''"■   """   '"   '' 'Ii"i'  (l!i'-lly.  Immmh-..   tlirv  iiMv  In.    iiM.>I  si,tUr;ict..nIv  n.iii- 

l""''''l    '•>    'I"'   ^"l"'-  "I'  til''  '•n:i>t;,n|   mC  |,,m|';iI|;|\   Iu   w'j  ilrli 'l  i|,.\     l.^ni. 

<  'liiiii.Uiiio'  the  loL:;irItliin>  t,.  luiiiii.ij  (jiLiniino  :iii<|  >lfvci(.|iiii--  in  |M.\\ri--  (.f  ir,  tin- 
;il)ii\('  i'.\|)i't's.si()ii  ;^i\cs: — 


>\\\  II 


l'~ 


I  )     \   -l-  r  ('(IS  / 


r'  I 


r  cos/  /  ^,.-  V 


;iiiil  tlii'ii 


,    SIM'/)    , 


III    (Ic\clu|)il|u'  ,     !•,,>    /    i^V((  lii(ll|ip(|s  of   cniiilillt.-llinli    wi  fc   llxd.   iis  ill   lllr  c  i|||  j  ill  t;i  t  inll 
"I'  til''   |irillci|i;il   Icrill   111'    llic   liititiplr. 

I.      I'Vdlll    (  '\\  I.I   \'>    IllMl-,    \\c    ||;|\,. 


(•(.s,/  — 


-1  .:■■ 


cos  4  ,: 


,   ''^^5    I 
+  r    ens 

:v^4 


;iii(l  then  l)y  siil)>titiitin^'  //   f-  //  '*> :  i\>v  :  \vc  Ii;i\c  ens  /■«lc\rln|)cil  In  iiiHl;i|i|cs  .,1"  //,  etc 

2.     I'llllill"- 


\vt'  Iiavf 


/~-.'/  +  'V 


ciis/'zz  ciis  '"^y  <'<>"i//  —  !^iii  '\/  sill  //. 


I) 


I  lie  \  nine  (il        w  ;i-  (1(  ri\til  li\-  1 1  a\S;-,n  iVoin  tlic  Icii;.;tli  ut'  tlic  >ccip]iil>  iiciidnlnni 
(I  ■  ■  ' 

Mini  the  (linicnsiiiiis  ut'  tlir  ciirili  iis  iuiniil  li\'  IIksskl.     The  ili-i'ix.itinn  i.■^  l:!^ i-ii  in  tiic 

A^lruiKiiii'isclif  Stichrirhliii,  \  ulunii'  X  \  II.  |);il;(.'  .VJ<-'.      'i'lic  <l;ila  in. hie  use  nl'  nn-:  — 

I),  rjidiiis  nl'cjirfli  inidi  r  llic  |i;i'  illd  aw  sin    s^' \     .       .      6370063  nicti'cs 

I*,  Iciin-Hi  ol'si'CdiHl^  ]i'Mi(liilnni  miller  s;iinc  jiiii'.illcl       .      O'''. 992666 

I 
hi,  mass  1)1   1I1C  iiicinn • 


78 

Tliti  rrsiili  is 


TRANSFORM  AlIiiN   Dl    ll\Ns|:NS  I.INAR    IIIF.OUY, 


I> 


liiy       zz  .'■'.J  I  71 II  ,U>. 


!Ir  j;iv('s  ii.--  the  rfsiiliiii:^'  (•(nisliint  pari  nl   liic  >:iic  ut  tlic  |iariillii\ 

iiml  tlic  cliaiiL.'!'^  Ill  fit''  ('oiistaiit  |)ruiliircil  li\   ->iiiall  cliaiiL^'o  in  tin'  data 


Iiicrcax'  (it  11""". I  III  I'  \ai-if>  tlic  cdii'^taiit  li\ 
liicrrasc  lit'  loiK)'"  in  |>  \ai'ics  the  (■iui>taiit  Ii\' 
liirrcasc  dl' iiiiitN'  ill  <lciiiiiiiiiiattir  III  //' 

rile  ili'Vi'luiiinciit  >iii)^rniiciitl\    'nvi'ii  jrails  to  a  ruii<taiit  i>t" 


—  O'.I  I 

4-0".  1 8 
+  o".i7 


3422  .09, 

a  result  (i".();  ;.''i'catfi' than  tiial  >taIiMl  li\-  IIaN'skN. 

1 11  (•iiiii|iariiiL:  iIm'  iiaraliaxo  nl  IIanskn  am!  I  >i;i,.m  \ai  tlii- milv  clcnirnt  wiiicli 
will  iiiatii'iall\-  all'iTt  the  i'r>iill  i^  tin'  (■Mii>taiit  ul  |iaralla\:  a  ciiniiiai'isiiii  nf  tlio 
(liricrciit  \aliii's  ni'  flii>  nut-taut,  wliicli  lia\r  liccii  rrci'iilU  ohtaiiiril,  will  tlirrcrDrc  be; 
of  intcri's:.  Tlirti'  ili>tiiirt  inclliods  of  ohtaiiiiiiLi'  this  impoi'tant  clfinciit  liaxc  iircii 
apjilicil 

((»■).  'I'hi'  thiiirclical  iiiftlioil  roiimlitl  mi  Kki'I.kk's  third  law  lis  cxpi'cssud  in  tlif 
thi'orN   111'  L:ra\  itatioii,  and  dt-rixi'd  rniidaniriitalK   Ironi  tlir  iMpiatioii 


II    II 


-'  =  m  +  >r, 


(I  iiriiii;'  the  iiicaii  distaiirc  ol  the  niooii.  uliirli  is  iiiiincdiatid \-  coninTtcd  with  the 
parallax:  //  thr  iiuaii  motion,  oj'  iIh-  \aliii'  of  wliirli  tin  re  is  110  doiilit,  and  ///  and  .M 
the  inassis  nl'thc  moon  and  earth,  exiires.-i'd  in  ainu'oiirial 


pnale  units,  the  ileteiinination  ol 


whiidi  is  the  nio>t  doiilitliil  part  of  the  iiroldeii 


(//).  .Measures  ol  the  moon's  po-ition  made  at  two  dislant  stations,  and  reduced 
to  a  eoiiimoii  moment. 

(r).  Meridian  derlinalioii>  of  the  iiioon  maile  al  the  same  station,  and  reduced  on 
the  li\potliesis  thai  the  midisliirlied  L!-eoeentric  oi'liit  is  a  i^reat  cinde. 

The  last  met  hod  is  not  w  ell  adapted  to  i;i\e  a  certain  result,  owiii;^-  to  the  constant 
errors  with  whiidi  inuasiires  ol'  alisohite  declinations  are  all'ected.  ^\'e  shall  tlierct'oro 
conliiie  our  consideration  to  the  lirst  two. 

'I'wo  deterininatioiis  l»y  method  (i-ri.  that  ol'  IIanskn,  just  (pmted,  and  that  ol' 
AiiAMs  in  the  Mniitlilji  Xnltcrs,  \'ol    XI 1 1,  and  the  llritish   Siiittkiil    A/hkuhic  I'or   1S56, 


aiv  a\ailaiile. 


Id. 


The  data  \\><cd  li\-  Mr.  Ada 


M.i  are 


I  >,  t'l'iim   liKssEi.,  and  iherelore  the  same  as  IIa.Nsk,\. 


1',*   ;,j-6   So  1 


(J  IMiLillsh  teet,  or 


///,  mass  III  moon, 


o"'.992  7i : 


81, 


piililisluil  jiaiiir 


riiis  \:iliu'  ill  Kiiglisli  led  wiis  Kiiiillvo.iiiiiiiniiiHlcil  hy  Mr.  Adamj,  liiniMir,  not  bcingoxiplicilly  nuotuil  in  IiIn 


TK.\NSI„„M.\ri..\  ..!■  I,  vNs,.N  ,s  ,  ,:n.\U  TIII.oRY 

with  IIa.n.skn  u..  |,,s„:_  .I^<IM„,,.^.:   ._,,,.      |  „  .•un,p;,n. 


79 


|iiiif  It 


Clmii-.-iii  |)-o,    -    -    -    cliin,-..  uf  .T„  _- o 
"       "    l'  =  -fo .040.      ••       ..        _,/,o5 


11       II    ' 


=   I-  '.5, 


•(  n 


+  o'  ,26 


^     •?' — —    iy    •'-    -    II->.:n-s  .,;„„.  ,1,1.   n..,l, ,1    ,.„.m...r 

;inn,MV  ;"";,'^^,-"'";:";':'''7^-  "•'•-•'••r-   I^Nsi.:Ns..n,i,nl. M,n„   ,„■  ,i,:„   .1.- 

'•';•'•    '•'•;'""■    ""•    -'l";>    '""1:  ..,•    il  .SH.:V  nu.l  A,.AMS  ,0   II  vvsKN's   .1.,,. nl 

iii^i  to  til.,  system  aliv,i,|\   .•H|u|,t,.,|,  i|„.  ,vs„lt,  ui||  !,,.;_ 

Cnll^tiiiit  (.iMiir  p;irillla.\,    1 1  AN>|.;.S.    ■^^2-".nq. 

"  "  11  » 

•^I'AMX.       ;i4^2    .13. 

('nllstailt  nl'  |.,irall,i\  i|,|.lt'.  IIanskv.   ;,.J-^2".2v 

"  Aham-.      vp:?  .2S. 

'I'iK-cn.istaiit  orn.(|„rtlni,  jmni  ,]„.  si,„.  ,,.  ,|„.  iMiallax  iisi.|f  is  4-,,     i^- 


//.    TIlc   llio.st    ITrcIlt   ilctcnililiatii.lis  nf 


llH'   III, .nils   jiarallas    l»\    iiicasiirciiici 


•  1  ,■    MI,  M  .  ,  |'Mi,,ii,,\     n\     lllcasii|',.ill(|i|     ;|i',> 

''•7'  "'   •^''•-  '""•■'•^-  '■^'" ''^  l-  A.  S.  X.XXIl,  „nl  „f  .Mr.  >p,.>..m!Ih,|    \\\IV) 

'""''  •"•'■  t'"iii.l,.,|  HI,  ( '..ij,,.  nliM-natioiis  ;,,„!  |„,tl.  lend  t..  a  .•.■ii>iai,l  of 


It  ,s  not  ilistnirtiy  statnl  wlu.tlH.r  this  i>  tl,..  ..unsfaiit  nf  ,|„.  parallax  Itself  n,  of  i^ 
sine  Ml'.  P.UKKN's  iiitrndurtinn  ,1.  ...  |,,,,  ,p,  ,,7.s,.,.,i..  to  iniplv  that  1,0  u.rA  Mr 
Adams  s  expn'ssion  l,,r  sn,..  |,,.,,allax  as  tl,..  parallax  itsril    i„  n-iliicini,'  thr  Cape  nl.sn- 

vati.ms.       Ilut.ii,  t|„.  ivihirti tl,.'  (iivMiuirl,  Ml,>MNatiM„>.  In    a  pplics  A  l>AM,s's  ,,,1- 

rcctioii  t(.  the  parallax  ,.f  Ann's  lii„ai'  ,v,li„tiM,,..  u|ii,.|,  nu,..  il,,.  p;in,llax  itself 
To  put  the  matte,-  into  a.iuihe,'  shape:  (  )„  p.  ,  ,,,  )\y,  D.^kkn  has  342j".33  us  the  con- 
stant of  parallax.  On  p.  i^j  he  h;,s  a  c.i.st;,,,,  ,,,ri,.etioii  of  o".r,S  to  the  Aiuv-I'lana 
parallax,  of  whirl,  the  constant  is  34.M  .So.  uhirh  j-ives  3422  .4.S  as  the  constant  of 
parallax. 

We  shall  probably  make  a  near  appi'.,xiniaiio„  to  the  truth  l,v  a.-^sni,,!,,-.  that  .Mr. 
r.KKKN'.H  mean  provisional  co,,.stant  was  3422  .40,  a,,.l  a.s  h,,.  (Ieiii,ce,|  a  co'nvciion  ,,f 
+  f"'  .38  tliLs  would  j4i\-e  IIS  his  result : — 


('oi,sta,,t  of  parallax,    . 
Constant  of  si,ie, 


-     3422".78 
3  42  2". 62 

Mr.   Stonk  also   linds  a   correction  of  +  o".3S  to  .Mr.    Ada.ms's   i)arallax.     This 
would  !^i\e  : — 

Cons[ai,t  ol"  paralhux,     . 
Constat, t  of  sii,e,      ... 


-      .      342  2".86 
.      .      3422". -o 

The  ovidcnco  is  thcrefoi'e  ii,  fasorof  a  positive  eoi'i-ection  to  IIanskn's  constant- 
hut,  in  accordance  with  the  practice  in  other  parts  of  this  paper,  the  results  as  printed 
are  all  founded  on  Uanskn's  fundamental  data. 


^ 


'J^ 


So 


TRANSKdRMA  I  li)\   oF   IIAS'SKN'S  IJ  NAR    rill.DRV. 


Ill   rllc  Tillilc   \'   till'  ciilllinllS  (•(ilil.lill- 
)         1 


(i).  The  viiliic  (if      •    -  -  .,.( I  +  ''  C's,/'),  cxprfsscd  in  sccoiiils  of  arc. 


)r. 


(2).   Till'  i»!'(Hli!cr  (if  tills  (|iiaiitity  Ky  —  /''  +   ,, 

(3).   Tile  (■(icliiciciils  fdi'  IIansi-.n's  sine  parallax,  fdniicil  Iiy  addiiiL;'  '  l)  ninl  (2). 
If  the  |iai'alla\    itself  is  rci|iiirc(l,  it  may  lie  fdiiiid  li\-  addiii,:;   flic   rcdiictidii  fniin 
tile  siiu-  to  tiie  parallax  iiscll',  iiaiiiciy  :  — 

-|-o".i5;     +o".oJ5  cos  // 

-f-  o".004  CDS  '//  —  2//'  A-  2  10  —  2  m') 
4-  o".oo4  cos  (  2  //  —  2  //'  +  J  <•'  2  m'). 

(4).  'I'iic  cdclliciciits  (if  !  »i:i.Ar\Av's  sine  parallax.  >o  far  .as  actually  coiiijuitcd  by 
liini.  A>  lie  stopped  at  the  tci'iiis  of  tlu'  fifth  order,  tlic  liiiii(lre(ltlis  of  scchhhIs  are  not 
always  dcfuntiN-c. 

(5).   The  >aiue,  with  tile  addition  of  (piaiitiiies  e>tiiiiatei|   hy  iiidiictioii    to    reprc 

sent  the  omitted  t(.'niis  of  liiLiher  orders. 

(6)  The  correctiniis  applied  in  the  precediiiL;'  colmmi  to  olitaiii  the  most  pi'olia- 
hle  \  allies  of  the  co<d'licients. 

(7)  'I'he  deviation  of'  IIansi.n's  ciwtliciei!t<  iVoiii  the  second  set  ol'  1  >i;lai'nav's. 
As  some  of   1  ))■,!. AlN'\\'s  teiiii.^   ai'"   doidifl'ill    tVom  the    iii-ufli'Mclif  coiivei'^.i'eiice  of 

his  series,  the  coeilicieiits  of'  .Voam-'s  parall.ax.  foimd  in  the  Mmithlii  Xcticcs  II.  A.  >S'., 
\'ol.  XIII,  p.  2<)'-,.  ha\c  iieeii  ad.ded  for  conrpari--on.  It  uill  he  seen  that  rhe\-  at^'reu 
(doselv  with  the  coeflicieiits  of   IIanskx.  thoiiuh  deri\cd  independently  of  llieiii. 


rRANsKORMATiux  oi-  ,,.^^s,:^■s  r.rxAR  TiiKoKv. 


8i 


4'     i- 


II, \z 

sin. 


CdS. 


sill  |.()S, 


sill. 


COS.  sin. 


-3      - 


+ 


+ 


<).  I  or) 

7.  "35 
7-73S 


I 

-3 

o 

-3 

I 

-3 

+ 
+ 

+ 


+■         (i.oii       _   U.O05 
+  11.075       —    o.i.i,^ 

*"        0.044      —  0.007 


o.oir      _ 
o.  240 

O.  T2() 
fl.0I2 


0.077 

o.S4f) 
o.-l.'^ 
o.  122 
o.oofi 

0.003 
0.0^; 


o .  00 1 


0.014 
0.003 


0.002 

0 .  002 


0.003 


—  t 

0 

+   0.003 

—    O.OOI 

0 

0 

— 

o.O(j4 

+    0.043 

—  0.010 

I 

0 

— 

2.524 

-{-    O.OS2 

-  0.03:; 

2 

0 

— 

0.052 

4-  0  <uo 

-  0.024 

3 

0 

—  0.002 

—   o.(,o4 

—  I 

—  I 

- 

0.040 

, 

+   0.001 

0 

-' 

+ 

3.665 

4-   2.324 

4-  0.046 

I 

—  1 

— 

27.620 

+  15-144 

+  0.0S5 

2 

—  I 

— 

23.00') 

r    S.4()9 

4-    0.045 

3 

—  I 

— 

'.337 

i-   o.(;S4 

f-    0.007 

4 

—  I 

— 

0.060 

-r    0.04I 

-2 

—  2 

— 

0.020 

--  0.003 

^' 

-2 

- 

i.S()3 

-  0.0S5 

^  0.007 

0 

-2 

— 

41.64^ 

+  "-3'.); 

4-   0,  762 

' 

—  2 

-i- 

4466.   icjj 

4-    I.S67 

+  2.3.SI 

2 

—  2 

+ 

2i44,,),j5 

■f-    I    4')l 

t-  1.750 

3 

—  2 

+ 

''10, 021 » 

•f-    r,.420 

-r    0.  |l  J 

4 

1 

+ 

2.0S3 

-    0.071 

t-   0 . 0 1 S 

5 

—  2 

•f 

0.0S4 

+    0.00  I 

6 

—  2 

+ 

0.004 

• 

• 

-I 

-3 

— 

0.0S2 

—    0.006 

0 

-3 

- 

2,3;:' 

-    '.437 

+   0.061 

' 

-3 

■f 

li)S.  103 

-    '4.S" 

*-   0.216 

2 

-3 

-r 

155. "47 

-    1) ,  1 40 

4-   0.202 

3 

-  1 

+ 

5.1 6(1 

-    1.341 

+   0.0511 

4 

-■  3 

4- 

0.  I(;S 

-   0.047 

4-  0.003 

5 

-3 

+- 

0 .  009 

—   0.00 I 

• 

O.  I  1.) 


+  o 

(-  o 

—  3 

—  o 


—  O 


.012 
.604 

■W) 
.1  =  2 

.  OOs 

.  I()2 
,  .,3s 
.1)05 
,  "22 
"75 
(,06 
11)5 
064 
043 


-f- 


—  0.004 

—  0.017 

-t-  0.010 

—  o.of.j  4 

—  (J.3)(i  -r 

—  0.036  - 

-t-   0.002 

-f-    0.0 ',s         -i- 

—  '•77'J       - 
4-   o.iif,       _ 

—  0.005 
4-  0.00') 
4-  0.27) 

f-  0.05S       — 
4-   0.003 

—  0,00a 

.       4- 


0.002 
0.004 
0.001 

(.1.003 
O..J2I 
0.071 
0.022 
o .  r)02 

0.004 
O  007 
o.t)02 


0.003 


(./,U)' 

COf. 


2 

I 

2 

-\- 

0.002 

0.000 
4-   o.O(j3 

( ) 

+ 

0.037 

—   0.039 

—    O.<i0I 

I 

"•351 

-   0.21U 

—   0.0  J  2 

2 

-4- 

0.127 

4-   0.012 

—   o.O(j6 

3 

4- 

O.OOI 

f-   0.003 

I 

- 

0.070 

""   "Y     " 

0 

+ 

..846 

—  o.2f.6 

—  0.024 

I 

— 

S5.224 

4-    1.633 

—  0.072 

2 

+ 

4.303 

+    0.941 

—  0.025 

3 
4 

4- 

o.O(j4 
o.oor 

4-    .1.073 
+-     '.003 

-t-    o.fV7I 

0 

-1 

i- 

0.04/^, 

-*-     O.OIjj 

—  O.OOI 

1 

-  I 

+ 

0.279 

+  0.295 

+  0.00; 

2 

—  1 

+ 

0. 1 19 

4-   0.077 

-•-  0.005 

! 

—  I 

4- 

0.003 

4-    0.0:3 

+     O.OOI 

1 

—  2 

4-   0.003 

, 

- 

—  2 

4- 

0 .  004 

4-   0.002 

■.">2 


TkANSFOR.MATIoN   nF   HANSEN'S  I.I  NAR  TIIICORY. 


TaiiM".   I. —  I'llhli'  (iJ'ik'^:,  (('■ 


-( 'oiitiiiiic 


.1  = 


("'':)■ 


f-   (1.007 


(/;.!:!' 


iiiA:.)' 


+ 


(1. 1114 


:'33 


3  '■'  -  3  '■' 

2  - 1 

3  -I 


3 

—  2 

4 

—  2 

0 

-3 

I 

-3 

2 

-3 

3 

-3 

4 

-3 

z 

-3 

1 

—  4 

2 

--I 

3 

-4 

1 

—  1 

0 

I 

-f- 

0 .  007 

+ 

O.OIO 

. 

I 

1 

- 

11.(131 

f- 

0.052 

-f 

0 . 0  j  I 

2 

1 

+ 

0.007 

• 

—  1 

{) 

r 

0.21)() 

— 

0.013 

-{- 

l.>.(-lo2 

0 

i» 

■i- 

0.31(1 

— 

0.01;.) 

+ 

0.017 

1 

( t 

+ 

17   SWj 

- 

0.  3S2 

.i- 

0.  OIlJ 

2 

( t 

+ 

I). 25') 

- 

0.044 

+ 

0.005 

—  I 

—  1 

_ 

11.  564 

— 

0.O2I 

— 

0 .  i.io; 

-   0.001 

1) 

~  1 

- 

1 1 .400 

- 

2.7"; 

- 

0.052 

—   0.002 

1 

-  1 

— 

121.33? 

- 

1.631 

- 

0.113 

-   0.002 

2 

-  I 

- 

I  .')iii 

- 

0. 163 

- 

0.042 

3 

-  I 

- 

o.'\37 

— 

0.012 

- 

0.    )OI 

—  1 

0 

— 

0 .  OOl) 

+- 

0.006 

- 

0.(JOl 

0 

-2 

- 

U.147 

+■ 

11-343 

- 

o.(,'i6 

I 

-;: 

— 

0.  5')2 

H 

0..176 

- 

0.014 

2 

-2 

— 

o.oSi 

t 

0.074 

- 

0.004 

3 

—  2 

- 

0.006 

+ 

0.004 

ij 

-3 

— 

0.007 

-t- 

0.012 

-*- 

0.001 

1 

-3 

-i- 

0  ii4[ 

+ 

u .  ( J 1  ■; 

-f- 

0.002 

3     -5      -f- 


+ 


+ 

O.OO.', 

+ 

0.001 

•r 

o.;;o2 

+ 

0.001 

— 

0.107 

0.03-* 

— 

O.O!  1 

- 

1.1.003 

0.  272 

0.410 

- 

0.014 

0.123 

— 

0.  2(J1 

— 

0.007 

0.00() 

- 

0 .  OOl) 

0.  (K)2 

^U 

l».Ot)*i 

+- 

(-'.001 

I  .OI)2 

+ 

0.2iO 

- 

0.044 

3.151 

-i- 

2.  70S 

- 

0.041)       +    ( 

).oo3 

0,621 

+ 

I.27.J 

- 

0.017        -^    ( 

).ooi 

o.otS 

-t- 

o.<i53 

- 

0. 1102 

0.001 

0.066 

+ 

0 . 0 1 S 

-I 

0.005 

0.22<J 

+ 

0. 150 

*- 

O.OI   ' 

0.07S 

^■ 

0   100 

-r 

0. 1)1 )',) 

0.003 

-U 

0 .  006 

+ 

0.  OOl 

0.012 

f 

0 .  00 1 

,            , 

0.007 

t- 

0.U04 

, 

.        .       4-   0.007  •         ■ 

—  0,001 

0.050       +   0.045       -t-   0.002 
0,757      —  <'.'M7      +   0.002 

—  0.C04 


,.,  +  1,1 


-3'' 


4,,/ 


—  5 
-5 

—  5 
-6 

-6 

—  6 
-6 

-7 


+ 

+ 


+ 
4 
4- 


rl  : 


0.0411 


(//.(:)- 


o .  007 


sin. 


(V,^3)^ 


0.024         —     0.001 


4-  o.(x)7 


—  1 

— 

f).002 

—  I 

+ 

0.037 

- 

o.o6i) 

—  1 

+- 

0.010 

— 

0  -  007 

~2 

— 

0.001 

^  1 

— 

o.tlo2 

. 

— 

0 .  tjua 

. 

4- 

0.002 

4- 

0.024 

— 

0.001 

0.324 

-(-• 

0.057 

+ 

O.OOI 

0  00; 

— 

0.004 

, 

0.033 

o.oi.S 


0.042 
0,350 
o,6oS 
0.236 
0.023 

0.026 
0.SS6 
30.040 
3:.  723 
10.6S3 
0.775 
0,04s 
o .  003 


0.017 

2 .  (>(,(, 

4  ■  1 4'  I 
1 .  5irt 
0.118 
o .  006 


:  u. 

0.2i)6 
o.  125 
0.010 

0.016 
o.ooS 


4-    0.004 


4-  o .  002 
4-  0.054 
-i-   0.023 

J-    0.002 

—  0.066 
4-  0.672 
t-    0.917 

+    0.323 

^-  0.02S 
+  0.036 

f    o..)24 

—  l7-'il<' 

—  46.370 

—  12.420 

—  0.665 

—  0.033 

—  0.002 

4-  o .  00  ( 
f-    0.07.) 

—  4-311 

—  ?  544 
~     1--^3J 

—  .>.  1  12 
~    0.006 

f      O.OO) 

—  0.254 

—  0.406 

—  0,15s 

—  0.010 

—  0.012 

—  0.022 

—  O.OOIJ 


4-  0.035 

4-  0.269 

4-  0.260 

■^  0.0S5 

-(-  0.010 

—  11.001 

+  0,013 

+  0.066 

-t-  0.077 

-r  0.035 

-t-  0.006 


0.020 
0.246 
0.259 
0.094 
0.013 

0.002 

o  026 
o  031 

0.013 

o .  002 

0.00 1 
0.0(,)2 


0.002 
0.001 


0.010 

0.039 
0.040 
o.oiS 
0,003 


0.001 

O .  (.KJ4 

0.007 
o .  004 


4-  O.OOI 


TRANsR,...M,MU)X,,FllANSl.:NM,rX.\K    lllKcuv. 


II /)Z 


I'OS.  si,, 


COS. 


•J-''- 

-  2  '-' 

M 

2 

—  I 

t- 

0.002 

—   0.0115 

—   o.ooS 

3 
4 

—  1 

4- 

o.oijS 

—  O.ooS 

—  0.002 

—  0.004 

I 

—  2 

-H 

0.015 

—  0.  14(1 

—   0.004 

- 

—  2 

— 

1  . 0(^2 

+    1.71/) 

■)-  0.024 

3 

—  2 

— 

0.002 

+   0.S05 

+•   0.026 

4 

—  2 

+ 

0.010 

-   0  017 

+  O.ooS 

' 

-3 

+ 

0 .  002 

—  O.ooS 

t-  0.001 

2 

-3 

— 

0.044 

+   <t.i>-(i 

+     O.OOf) 

3 

-3 

— 

0.041 

+   0.05S 

-t-   O.ooS 

4 

-3 

+ 

<i.O(j2 

+  0.002 

O 

-4 

+  0.003 

3 

-4 

■ 

+  o.(x)3 

2  (-)  — 

O 

--3 

—  0.001 

+     O.OOI 

' 

-3 

— 

0.023 

-t-  0.05S 

+  0.009 

2 

-1 

— 

0.012 

+  0.027 

-f    0  004 

(> 

-4 

-t- 

0.020 

—   o.oSd 

+■  0.001 

I 

--4 

+ 

0.214 

-    •■^34 

—    <">.02S 

2 

-4 

-I- 

0.22.8 

—  0.628 

—  0.025 

3 

-4 

- 

0 .  o6f) 

-(-    0 .  uyo 

—  0.006 

•1 

—  4 

— 

0 .  005 

+   0.004 

0 

-5 

• 

—   o.oof) 

—   o.uoi 

I 

—  5 

+ 

0.021 

—    0.1(j2 

—  0,011 

2 

- ; 

-1- 

0.030 

—   0.07a 

—  0.007 

3 

-5 

o.ooS 

+  0.010 
—  O.ooS 

—   0.002 

I 

-f. 

2 

-6 

—  o.oof) 

. 

5  '•'  -  5  <■' 


-4 
-4 
-4 

—  5 

—  5 
~5 
~"5 
-6 

—  6 
-6 
-6 


COS.  sill. 


I'OS. 


■  (,<.<■ 


— 

O.OOC) 

— 

0 .  006 

• 

— 

0.010 

— 

0  006 

• 

- 

0.002 

- 

O.OOI 

+ 

0.  026 

-■(- 

0.003 

t      O.OOI 

0.056 

+ 

0.0()|) 

f- 

0.045 

f     O.OOI 

O.IK); 

+ 

0.042 

t- 

0.042 

0.007 

— 

O.OOI 

- 

O.OI  1 

. 

+ 

0.002 

, 

, 

0.006 

+ 

0.012 

f 

o.c)04 

,            ^ 

0.001 

+ 

0.003 

+ 

0.005 

0.0(J2 

+■ 

O.OOI 

- 

-  5 

—    0.002 

—    O.OOI 

3 

—  5 

— 

0.004 

r-    0.01  1 

f     O.OI 1 



0.003 

4 

5 

6 

-5 
—  5 

- 

0.01  I 
O.ooS 

-r    O.02S 
+    O.OlS 
+    0,004 

+  0.022 
+  0.01, 

■(-   o.(:o2 

- 

0.005 
0.0:  .3 

2 

-6 

■T 

0.009 

—    0.013 

+   0.015 

.1 

-6 

+ 

0.2S5 

-    0.652 

—    0,50s 

4 

—  6 

+ 

(J,  53S 

—     I.0S2 

-  o-ri'j 

5 

—6 

-f- 

'5-33  4 

—    0 .  6 1  M 

—  0.35 1 

6 

-6 

4- 

0.034 

—  0. 13S 

—  0.079 

/ 

-6 

+- 

0 .  009 

—  0.012 

-■  0.006 
+  0.002 

] 

- 

—  0.001 

3 

-7 

-t- 

0.037 

—   O.oSS 

--  0.070 

+ 

0  003 

^ 

/ 

+ 

0.0S5 

—   0. 176 

--  0.126 

-^- 

0.005 

6 

/ 

4- 

0.061 

—   0.116 

—  0.076 

-i.- 

0.003 

/ 

-(- 

0.016 

-   0.029 

—    'l.OI- 

/ 

/ 

• 

—   0.002 

—  0.001 

3 

—  s 

—   0.007 

—  o.f/05 

4 

-S 

. 

—   0.016  ■ 

—  0.012 

5 
6 

-s 

-8 

—  0 .  0 !  1 

—  0.002 

—  0.007 

—  O.OOI 

-4 
-4 


4,.. -61,/ 

1  -6 

2  —6 

3  -(> 

4  -6 

:      -6 


3     -7 


• 

— 

O.OOI 

— 

0 .  002 

O.OII 

-t- 

0.035 

+ 

0.02S 

0.016 

■t 

0.03S 

+ 

0.027 

0.005 

-r 

0 .  (JO<) 

+ 

0.006 

• 

+- 

0.003 

- 

0.003 

• 

t- 

0.005 

4- 

0 .  003 

— 

O.OOI 

— 

O.OOI 

0 

003 

- 

0.017 

-- 

0.030 

0 

005 

- 

0.019 

— 

0.025 

0 

001 

- 

0 .  004 

- 

0 .  o(  >4 

• 

+ 

0.002 

+ 

0 .  00 1 

- 

0.002 

- 

0.004 

- 

0 .  o(j4 

- 

0.004 

84 


TKANSFOR.MATKiN  OF  IIANSI.N'S   I.INAK    I'l  I  i:t  iF<  V, 


'Alll.r.    II. —  I'lhir't  1,1(1  jKirls    iif  IIaXSKn's     /\r//ji/ir  I.iiliilihlili\   ir'llli  the  Cnrljiciciih   (if  / III' 

('iii'(ii((l((l  Liii/i/ih((l(. 


('•■A')i 


(lis  X 


iS> 


(«.';)•  X 


{iii\z)'X 


('■•A'h 


Suiii. 


'•J    ^ 


1 1'liiis  in        — . 

iiplir  I, oil-     .S-  ■ 

,nilii<li',  .5 


-    .16 


-4     - 


3      - 


'      -3 
2      -3 


2 '.)  —  ; 

—  I 
() 
I 
2 

3 

—  2 

—  t 


3 
4 

(i 
-  1 
-3 


—      .  ij  1 1 ) 


.253      — 


003 


-i- 

.1138 

.  1111 1 

r- 

•  =  '? 

.012 

+ 

fi.?25 

.022 

H- 

3('-Sf>3 

•  ■■^T 

-t- 

10. 157 

•  "34 

-f- 

:,(...|2; 

.ooy 

+ 

S.'i2; 

.iK-2 

^ 

.f.65 

t     + 

.041) 

+ 

.  "03 

+ 

.011; 
■  *  7- 

-i  ■ 

.410 

.027 

-r 

.103 

. 

+ 

.407 

. 

-(- 

.I4f' 

. 

+ 

.012 

, 

+ 

.(CI 

+ 

.004 

. 

-t- 

.003 

-f- 

.C04 

+ 

,(.03 



.015 

- 

.13.1 

.i"il 

— 

.(  iiS 

.fOd 

— 

■  1 3') 

a- 

.013 

.  004 

4- 

.0.^1 

'"4 

— 

I .f  (15 

rj(, 

~ 

1  .1  ''7 

)'I4 

- 

1 .  5711 

l'(;2 

- 

i  ■:-<") 

0()() 

- 

.  If  s 

- 

.<ir  5 

- 

.DOI 

-t- 

■ '  '5 

4- 

.071 

00  [ 

4- 

.i^.Ni 

2'''3      -   3-41)3 
'44      -    i-7('fi 


- .  1-11-13 
-  .012 
.oil'!       -r 


+  .001 


-)- 

.001 

+ 

.ooS 

1- 

.063 

+- 

.K2 

~ 

•  ■'45 

~ 

•'43 

— 

.107 

— 

.oiS 

— 

.002 

— 

.002 

— 

.  OOI| 

— 

.  026 

— 

.002 

+ 

.026 

+ 

.01 1 

+ 

.002 

+ 

.(K)3 

— 

.002 

— 

.003 

+• 

.014 

4- 

.12^) 

-U 

•  4?7 

+  .  002 

4- 

■'73 

+  •"37 

- 

•  4'>5 

-I-  .010 

— 

•  292 

J-  .DOI 

- 

■  "f'S 

- 

.00() 

_ 

.001 

•  52f' 


-         2  •  !<07 


004 

007 

uof) 

(107 

no 

00 


4- 

+ 


+ 
4- 
4- 
4- 

4- 
4- 
4- 

+ 
4- 
4- 


4- 
4- 

+ 


4  22(137.  150 
4-  7fiS..'!;S 
,619  4-  36.  1 12 
.0(15       4-  1.1)31 

.005      4-  .113 

007 
•  003 
• <'3>) 
•551 
7 .  f)f)h 
I0().qi)8 
6611.852 
I  4  S  ,  I  )oo 

')-7"J 

670 

.047 
.003 
.003 
.065 

1.1^4 

7.5"7 

.192 
.014 
.00: 
.01? 
.078 
.04S 
.003 


•  039 
.522 

I  m'h) 
3(1 . 1  •'4 
12.3^. 
36 .311) 

S.J04 

•644 

•  047 
.(103 
.003 
.063 

•3S4 
.18S 
.434 

•  '57 

•  014 
.001 
.004 
.003 
.004 
.003 


.015 

•  136 
.01 1 
.136 
.013 
.oiS 

•  217 
'•'44 

•  031.1 
K446 
i.SSf) 

.226 
.024 
.002 
.005 
.071 

■975 


4- 
4- 
+ 
4- 
4- 
+ 


4 

4 
4- 
4- 
4- 
4- 
4- 
1- 
4- 
4- 


+ 
4- 
4- 


.015 
.230 

2  •535 
•  iSS 
.013 
.oiS 
•177 

2.^21 

28. 559 

24.452 

2 .  926 
.292 
.021 
.  002 
.005 
.071 

•949 


e  e 
e 

t  e 


TRANSFORMATION  OF  MAXSKX'S  I.rNAR  T,„:oRV. 


8; 


« t! ;  X 


s    s 


2  M  - 

—  I 

() 

2 

3 
4 

5 

r. 


—  I 
() 
I 

2 

3 
4 

5 
6 

—  I 
o 
r 

3 
4 


—  3 

-3 

-3 

-3 

-3 

-3 

-3 

-3 

-3 

-3 

-4 

-4 

-4 

-4 

-4 

-4 

-4 

-4 


.rS„ 


—  .013 
+  .137 
+   3-3<M 

—  23.322 

—  2.653 

—  .246 

—  .021 

—  .  002 


(« -'  :)•■  X 


I'-,  .,'1. 


.001 
.01)4 
.  1 50 
1 .001) 
.126 
.1)12 


.004 

•"33 
.004 


+ 
+ 

4- 

+ 

4- 
+ 

+ 
4- 
-f- 

+^ 
-4- 
-t- 
+ 


+ 

+ 
4- 

+ 
+ 
4- 
+ 
4- 
4- 
4- 


+    253.133      4-    .003 


115.652 
24S.2<)3 

'34.633 
1  2 .  561) 

•  C77 


•3a3 
.•J<8 

"•453 

S.3.j() 
II. 147 
9.265 
.921 
.076 
.006 
.002 
.023 

•  415 

•  4l'J 
.402 
•452 
.047 
.004 
.013 
.oiS 
.014 

.  r)  1 9 
.  002 


.166 
.006 
.  00 1 


('•..V!: 


—  .Oil 

-1-.003 

4-.  01 6 
— .  003 

—  .  1 11  )6 

—  .  00 1 


-.035 

4- 

—  .IJI  I 

+ 

—  .(«J2 

4- 

+ 

4- 
4- 

+- 


4- 
I- 


('"'.:)■' X 


('•.iMi 


.019 

—  .001 

-  .  t)  1  , 

•"5'} 

~.(04 

—  .026 

.032 

— .  002 

-.024 

.041 

— .  004 

-  .02S 

.046 

-.U03 

—  .021 

.oiS 

—  .001 

—  .ooS 

.  OrX) 

—  .oni 

.  '  II  I  [ 

•       • 

.012 

.09') 

—  .oor 

•  44.' 

—  .002 

.217 

-.003 

•377 

—  .001 

~  .  Oi;2 

.306 

-•«M 

.073 

—  .001 

.01 1 

•• 

.001 

.on; 

.026 

.01  i 

.  02i) 

.023 

.  006 

•      ■ 

.001 

.Dili 

.  00  [ 

.f  III 

+ 

4- 

4- 


I- 
4- 

+ 
4- 


1 1'Miis  in        ^   -z 
l'"i  liiilii- 1,(111-     —  ■- 


4- 


15.082 

253^3<'5 
1 1 S. 962 
224.748 
131.S99 
12.291 
.976 

•075 
.005 

.0(12 

•031 

•  5^7 

1 1 .012 

S.329 

10.470 

'}-43r 

.9S4 

.0S7 

.007 

.  002 

.oiS 

•  3S9 

•  405 
•3''7 
•471 
.053 
.004 
.012 
.017 
.015 

.  <J20 
.002 


4- 


-f- 
4- 
+- 


4- 
+ 
+ 
4- 
4- 
4- 
+ 
+ 
4- 
4- 
+ 

4- 


13.1S9 
211.655 

45S5^054 

2369.746 

I'll  ,921 

'4. 374 

1 .060 
.07.) 

.(M5 

.002 
.031 

•475 

S,66o 

206.432 

'^15.517 

'4  •597 

I.1S2 

.096 

.  00  7 

.002 

.oiS 

.  2S0 

7-440 


...64 
.004 
.012 
•  257 
•344 
■  032 
.002 


4- 
4- 


4- 


.002 
.011 
.001 
.010 
.oS, 
.054 
.115 
.010 


.iSo 
.  ()26 

.053 

.04,) 

.007 


4- 
4- 


.019 

.  00(J 

.019 
.  006 


•  C'03 

.(lOI 

4.657 

.55S 

4.650 

.084 

.002 


4-  .004 
4-  .015 
4-  .  004 
—  .010 
4-  .  152 

+  1.341 
4-  .366 
4-  .052 
4-  ,  006 


.009 
.  002 


.001) 
,002 


4-  .002 

—  .002 
4-. Oil 
i-.oo4 
4-  no  I 


4-.  001 
4-  .015 
4-  263 
4- .  1 30 
4-.  005 
—  .(JO  I 


.00  1 
.002 
.  006 
.(jor 


.001 

.  0(  I  [ 

.04) 

.  03  t 
.042 
.032 
.  006 
.001 


.00! 
.002 

.  OU I 

.0(j6 
.012 
.005 
,001 


4-. 
4-. 
4-. 
4-. 


002 
0S7 
264 
158 
"35 
008 
001 


4- 
4- 
4- 

4- 
f 
4- 
4- 
4- 
4- 
+ 


4- 
4- 
4- 
4- 


.  005 
.004 
.002 
.027 
.  264 
.  2()0 
.264 

.  "43 
.006 
.  002 
.001 
3.424 
.014 

4-351 
.  096 
.042 
.  (xnj 
.  00 1 


4- 
4- 

4- 

4- 

4- 
4- 
4- 
4- 
4- 
4- 
4- 


.005 
.006 
.002 
.010 
.0S7 
.416 
.265 

.043 
.006  i 
.002  ' 
.065 
I.0S3 

39.5*3 
411.674 

45.091 

3.'W7 
.329 

.02t) 


86 


transfokmati'jn  or  iiansens  ij-nak  tiii:ory. 
Taiilk  II. —  Thi'  MiiDii's  Loi/fjiliidc — (yOiitiuiH'd. 


J>o 


('■■A')i  -  I 


>n\:  X 


Ri.i 


s  Sj 


(>n^z)-X 


U-.K)' 


[n  (!  ;f  X 


(■■.  .di 


T<'riiis  ill        —    c 

Kcliplic  I.on-     .Et'c 

giliido.  .5  '% 


I 

—  I 

+ 

004 

4- 

001 

.   .1 
i 

4- 

.005 

4- 

.005 

. 

o 

—  I 

— 

(tf)7 

+ 

Olf) 

4- 

.  oof) 

4-  .002 

4- 

ooS 

.   .  1   .   . 

4- 

.025 

4- 

.071 

1 

—  I 

— 

144 

4- 

(lOI) 

- 

.074 

+  .00; 

4- 

«)S 

4- .  (X)3 

- 

.11)') 

4- 

.oSo 

2 

—  1 

+  I 

lO) 

4- 

016 

-1 

.324 

4- .  005 

- 

(MJ( 

4-. 010 

- 

.l(,S 

- 

.07S 

3 

—  I 

+   . 

123 

4- 

008 

— 

.444 

—  .001 

— 

002 

1- .  OOt) 

— 

.307 

— 

.304 

A 

—  I 

+   . 

OK) 

- 

.065 

.   . 

. 

4- .  002 

- 

.053 

- 

■053 

5 

—  I 

— 

.007 

. 

— 

.007 

— 

.007 

I 

—  2 

— 

Of)  2 

- 

.001 

. 

- 

.(X)3 

- 

.003 

2 

__2 

+ 

Olf) 

— 

.015 

-1-  .  002 

. 

—  .002 

-f- 

.001 

4- 

.005 

3 

2 

+ 

003 

• 

.(X)7 

•   ■ 

—  .001 

.  005 

~" 

.005   . 

2  ( 

4 



007 

,  , 

_ 

.007 



.007 

0 

4 

-)-   . 

"34 

4- 

.01; 

—  .001 

—  .  002 

4- 

.  046 

— 

.068 

I 

4 

- 

007 

4- 

.017 

. 

—  .OUI 

-4- 

.009 

4- 

.  009 

2 

4 

4- 

.002 

4- 

.002 

4- 

.002 

3 

3 

. 

+ 

002 

. 

4- 

.002 

-t- 

.002 

2 

3 

4- 

020 

. 

, 

— 

00.| 

.   . 

4- 

.Olf) 

4- 

.028 

I 

3 

+       ■ 

014 

— 

190 

- 

.  oof) 

—  .ooS 

- 

010   —.002 

- 

.202 

1 

.  402 

O 

3 

-+-  I. 

032 

+ 

025 

4- 

.283 

—  .032 

4- 

001 

-.oiS 

4- 

I  .  296 

— 

2.153   '' 

I 

1 

— 

024 

— 

1S7 

4- 

.44f) 

+ .  oofi 

4- 

010   —.035 

4- 

.216 

4- 

.064 

n 

3 

—   . 

ooS 

— 

022 

4- 

.  0f)f) 

4-. 001 

4- 

003    —.011 

4- 

.  029 

-t- 

.029 

3 

3 

_ 

002 

4- 

.ooS 

—  .002 

4- 

.  004 

-f- 

.004 

4 

2 

, 

+ 

002 

. 

. 

4- 

.002 

4- 

.002 

3 

2 

+-   . 

001 

4- 

023 

- 

003 

4- 

.026 

4- 

.031 

o 

2 

+   . 

ooS 

4- 

2.JI 

- 

.003 

—  .ooj 

- 

02S   ,  . 

4- 

.2f)3 

4- 

.425 

I 

2 

-t- 

ng4 

-   4 

504 

- 

.117 

—  .UOI 

- 

040   .  . 

- 

4-577 

4- 

&.3f,3   I- 

(.1 

2 

+  23- 

()()0 

4- 

342 

4-3 

.794 

4-. 016 

4- 

024   4-.  003 

4- 

27.839 

— 

55.262   I- 

I 

n 

-   . 

(.76 

—   4 

450 

+  <) 

.62S 

4- 

0=2   4-.  005 

4- 

4.551 

— 

•  175 

2 

2 

— 

275 

— 

5()4 

4-1 

.472 

4- 

003   +  ,  002 

-r 

.()3S 

4- 

.561   . 

3 

2 

- 

02  S 

- 

044 

4- 

.167 

•  • 

4- 

.095 

+ 

.095 

4 

2 

— 

003 

- 

003 

4- 

.OI? 

.  .  1 

4- 

.012 

4- 

.012 

5 

2 

4- 

.002 

. 

.   *    . 

4- 

.002 

4- 

.002 

2 

— 

003 

. 

4- 

00 

.   . 

— 

.002 

— 

.008 

1 

— 

oOvS 

+ 

113 

4- .  iiuf) 

4- 

(JO 

■   4-. 002 

4- 

.  I20 

__ 

.07? 

o 

- 

5"5 

— 

013 

- 

•  044 

-^.034 

4- 

0<J2   4. Olf) 

- 

.510 

4- 

1-554   ' 

1 

— 

020 

4- 

113 

— 

.of)3 

—  .003 

— 

ooS   +.035 

■+• 

.052 

4- 

.cog  '  . 

2 

- 

003 

4. 

oof) 

- 

.(J03 

—  .002 

- 

003   4.013 

4- 

.of)S 

4- 

,008 

3 

4- 

001 

4  .  002 

4- 

.003 

4- 

.003 

o 

O 

- 

0(  ■'() 

■   •     . 

— 

.(X)f) 

+ 

.006  i 

■ 

O 

• 

- 

.005 

.   . 

- 

.005 

- 

.005 

, 

(.1  +  2  (j' 

2 

3 

- 

003 

4- 

.(X)7 

•   •  1 

4- 

.004 

4- 

.004 

• 

3 

3 

4- 

.002 

. 

4- 

.002 

4- 

.001 

o 

2 

. 

4- 

001 

— 

.  ( )0 1 

+  .oor 

4- 

.001 

4- 

.001  !  . 

I 

2 

+ 

ni6 

. 

- 

.031 

—  .  o(J4 

—  .002     .  • 

- 

.021 

— 

•"34   • 

2 

2 

— 

oSl 

— 

<WI 

4- 

.  If)0 

-  .  ix)4 

4- 

.074 

4- 

."75  \     • 

3 

2 

— 

oo3 

4- 

.036 

—  .001 

4- 

.027 

4- 

.007 

4 

2 

• 

4- 

.005 

. 

4- 

.005 

000  \     , 

2 

I 

+ 

001 



— 

.004 

t 
.... 

I 

.003 

— 

.003  1  . 

TKANSFCRMATION  o[    IIANSF.N'S  I.INAR  TIIKORV. 

Faulk  1  ]  —The  Mnnns  l.„u,i]tn(h~(  'oisdukmI. 


«; 


n<^zK 

('"-"=>jX              («-l:fX 

■ 

■ 

4 

s 

A'' 

-' 

s,, 

*} 

Sum. 

i-:c 

■fiins  in 
ipiic  l.oii- 

^  ^ 

u- 

-0)' 

(<•. 

,c)i-i          Ri,, 

/S, 

( 

'>:■^^          (Ri,:)         {e,g\. 

Hilu.k: 

i 

'' 

•»■ 

,,                ,, 

■\  : 

I 

~~ 

.(X>2 

•  ,       ■■   • 

-f- 

.<302 

• 

.000 

+ 

.007 

2 

I 

_ 

,()('2 

• 

— 

.031 

—  2 

() 

+ 

.022 

.0.12 

— 

.  o<  l.( 

- 

.004 

—  1 
o 

1 

2 

3 

o 
<) 
o 

0 

o 

+ 

.006 

4- 
4- 
+ 
4- 

4- 

•"S3 

•"33 
•  '/'5 
.oSo 

-*- 

.01)4 
.010 
.'X)2 
.Oil 

4- 
.        + 

.      4-. 
.      + 

.022 

.O-Q 

•977 

•  "35 

•  97f> 

4- 
4- 
4- 
f 

-f- 

.022 

•  3^'9      ~ 
1 .  2'.I3      -  >; ' 
I7.''"I       T,-' 
J  .235      r  ec 

4 

o 

4- 

.006 

■ 

+• 

.003 

.      4- 

.083 

4- 

•  "■S3 

'   -3 

.      4- 

.  oof) 

+ 

.  o(j6 

—  I 

— 

.wjS 

• 

_ 

.(XJI 

'      1 

—  I 

.107 

.  0(.»9 

.  oog 

—  I 

1 

■ 

~~ 

.on 

— 

.llS 

„ 

.IlS 

—  I 

1.053 

- 

.082 

1 .  16; 

I    "''(1 

1       ° 
I 

—  I 

— 

."5S 

— 

O.OqC 

- 

■045 

•       4- 

,  00 1      — 

6.7()S 

_ 

.  ],' 

—  I 

— 

■"47 

— 

•  717       +    .<-■( 

2     — .  'J07 

4- 

.07 1 

2 

3 
4 

() .  704 
•549 

-\- 

.  00 1      — 

•  ^")7 

— 

122.032         T 

—  I 

—  I 

+ 

+ 

.<)2(J 
.002 

- 

—    'JCI 

4- 

.055 
.012 

.            -f 

001      — 

6.630 
•  53; 

- 

S.24'j      r 
■  572 

(' 

—  I 

.oil 

4- 

.002 

5 

—  I 

— 

.IW3 

■"39 

"" 

•"39 

n 

_2 

- 

.  00 1 

• 

4- 

.002 

-v 

.003 
.oor 

4- 

.  003 
.  00 1 

o 

" 

.010 

.    . 

4. 

.o<J7 



.003 

.012 

** 

~ 

.UOl 

.032 

•    • 

-1- 

.013 



.  020 



•  "'7 

I 

- 

— 

.1)01 

.013 

. 

— 

,  ijoS 

' 

.1)22 

.5S4      - 

e 

3 

—  2 

I 

.032 
•""7 

—  .-JOI 

.015 
.004 

- 

.046 

- 

.127 

.011 

.017 

() 

-3 

4- 

.i")2 

J- 

.ix)r 

I 

-3 
-3 

4- 

.003 

.004 

1 

1 

2 

4- 

■  >M2   ,. 

*        " 

- 

.IXJI 
.(X)I 

+• 

.  00 1 
.001 

+ 

.040 

.0(JI 

3"- 

■3„/ 

() 

• —  2 

— 

.001 

' 

.002 

,      — 

.003 

— , 

.  003 

I 

—  n 

4- 

.01(1 

-    - 

— 

.013 

4- 

.003 

_ 

•  035 

'         2 

rj 

+ 

.00; 

—  .lint 

— 

.  oof  1 

.002 

H" 

.270 

!       3 

—  2 

~ 

(){)I 

+ 

.i)i6 

-       . 

-j- 

.013 

4- 

.'127 

+ 

.  1 5" 
.024 

4 

—  o 

+ 

.of.>S            .     . 

•       ■ 

4- 

.007 

4- 

.015 

+ 

5 

—  2 

4- 

.001 

■       . 

-1- 

.0(51 

4- 

.002 

^- 

.002 

i   -' 

-3 

• 

— 

•"05            .     . 

-       • 

4- 

.002 

_ 

.  "03 

— 

.003 

o 

-3 

• 

~" 

■■'72            .     , 

•       • 

■»• 

■  "1 7 

— 

■  "55 

__ 

■"57 

' 

-3 

— 

001 

— 

•  '71           .     . 

— 

.oSo 



.0()2 

. 

••154      - 

2 

-3 

-f- 

(>()() 

■— 

.02f.               .       . 

—  .fyrn 

^ 

.02') 

-V 

.01  I 



3-'13      - 

e 

3 

1 

—  3 

+ 

01  S 

~ 

.176         .    . 

~.rf^, 

— 

■"74 

— 

.226 

■f 

■395      ^ 

!       4 

-3 

+ 

<"'3 

+ 

.0:2          .     . 

—  .-,"0  1 

— 

■  "45 

— 

.UI.) 

— 

.1  01 

•       5 

--3 

4- 

.003 

.      - 

— 

.ooS 

— 

,005 

__ 

.  004 

;      o 

-4 

•    i 

— 

.005 

-      - 

4- 

!»! 

— 

.  00  ( 

— 

.  004 

' 

—  ) 

— 

"13         •    ■ 

-      - 

4- 

.005 

— 

.ooS 

_ 

■  "74 

;   == 

-4 

+ 

."01                .      . 

.      - 

4- 

.  fXJ2 

4- 

.  003 

_ 

.  226 

i       3 

-4 

— 

.013        .    . 

•      - 

— 

.004 

{       • 

— 

.017 

-f 

.061 

i    ♦ 

-4 

+ 

.004          .     . 

.      - 

— 

.003 

+ 

,001 

4- 

.004 

!         2 

~5 

*       '                *       ' 

* 

— 

.012 

3 

—  5 

.001          .    . 

• 

• 

— 

.001 

4- 

.(X)6 

88 


TRANsroiorA  ricN  oi"  Hansen  s  ijnak  theory. 
'rAiii-i:  II. —  /'//'•  Mniiii's  Ldi/i/ihii/i — ( Vi]itiiiii<-«i. 


.T 

A 

s 

s. 

« 1' :  X 

Ui<\: 

•X 

Kniz^-K 

Sum. 

T 

E.:li 

urms  ill 
ptic  u,  :i- 

i 

C 

(',.<). 

—  I 

K„, 

fS, 

(<■ 

.<) 

K,.s 

it,t\t 

fi 

ituilf. 

£ 

1 

rj  -f-  LI 

,, 

,f 

,, 

,, 

,, 

2 

2 

+ 

.rx)r 

, 

.     *, 

. 

+  .U 

I         .     , 

- 

.002 

4- 

.fA)2 

. 

1 

-I 

,        , 

+ 

.  01  >6 

.       + 

.001 

.     . 

- 

.007 

4- 

.  007 

O 

— 

.02) 

+ 

.042 

4- 

.010 

— 

.001      — 

.002 

- 

.023 

4- 

.073 

, 

I 

+ 

.Ot-l 

4- 

.  o(  )3 

— 

•  243 

- 

.006      — 

.001 

—  .0( 

3         -     - 

- 

.lift 

4- 

.541 

. 

2 

+ 

•"3? 

+ 

.042 

- 

.063 

+ 

.002 

— . ooO         .     . 

-r 

.010 

+ 

.010 

, 

3 

4- 

."04 

+ 

.01 '3 

- 

.010 

—  .00 

1 

- 

.003 

- 

.0(55 

, 

—  I 

0 

,      , 

4- 

..""3 

-r- 

.003 

4- 

.003 

. 

0 

I 

0 
0 

•     • 

+ 

+ 

.002 
.(103 

+ 

.002 
.035 

—  .IX 

I                   .          • 

■#- 

.000 

.037 

4- 
+- 

■049 
.061 

■ 

2 

0 

- 

.003 

+ 

.  0(I2 

4- 

.006 

•       • 

-»• 

.005 

4- 

.005 

:        . 

3''' 

-,..' 

! 

2 

0 

-r- 

.003 

, 

— 

.(^02 

• 

-- 

.0«>l 

+- 

.001 

, 

3 

0 

- 

.035 

- 

.uol            .       . 

— 

.')3fi 

- 

.036 

. 

4 

0 

— 

007 

.       ■    . 

— 

.007 

— 

.007 

I 

—  I 

-^ 

f)03 

4- 

Ot)2 

— 

.(K)2 

.           . 

^ 

.003 

4- 

.oo-^ 

. 

2 

—  ! 

— 

(125 

■i- 

OOI) 

— 

."<y<      .    . 

— 

.022 

4- 

.015 

3 

—  I 

- 

IlIO 

4- 

002 

4- 

240 

4- 

.U02 

- 

.00 

■                   *          * 

■s- 

-23  = 

4- 

•  245 

. 

4 

-  I 

- 

(iOl 

+ 

001 

4- 

"43 

- 

.001       .    . 

*■ 

.042 

4- 

.042 

: 

—  I 

4 

005 

•        • 

.     . 

~ 

.005 

4- 

.003 

. 

3 

—  2 

— 

001 

• 

4- 

(JO  I 

-i- 

.001 

-r 

.001 

4. 

.OOI 

. 

Li  — 

y.: 

—  I 

—  3 

- 

C^OI 

- 

toi 

4- 

OOI 

-i- 

0<JI              .        . 

- 

.004 

4- 

.004 

, 

• 

0 

-3 

-r 

1)1 1 

- 

DiS 

4- 

007 

4- 

.002 

— 

IXI 

5         -     - 

— 

.003 

— 

.003 

, 

1 

-3 

T- 

017 

— 

DO  2 

- 

.oo( 

- 

OJ3         .     . 

- 

.oc6 

— 

.3IS 

, 

2 

-3 

. 

- 

oiS 

4- 

cot 

- 

.'»2 

•     - 

- 

.019 

- 

.014 

, 

~ 

-3 

— 

001 

.         .     . 

— 

.001 

— 

.001 

. 

41..- 

I 

4'-' 

_ 

0<52 

+ 

.002 

.'JiXl 

.000 

2 

—  2 

— 

001 

•1- 

.001 

. 

.a» 

— 

•"33 

. 

3 

2 

— 

0')2 

— 

.002 

. 

— 

.'*-i\ 

— 

.022 

^ 

4 

__  2 

- 

001 

— 

.001 

. 

— 

.>T02 

— 

.002 

, 

0 
I 

3 
—  3 

, 

-u 

001 
022 

+ 

.001 
.022 

— .  '.<:■% 

■K 

.003 

4- 
4- 

.002 
•039 

2 

-3 

. 

— 

"3- 

• 

4- 

.02S 

—    "'-2 

— 

.006 

— 

.356 

, 

3 

-3 

-4- 

007 

— 

032 

4- 

.<x)7 

— 

.010 

—  .V>i 

- 

.032 

- 

.(140 

i'  I' 

4 

-3 

-r 

005 

— 

037 

+ 

■  o'j; 

— 

.027 

-.005 

— 

■'^37 

- 

■293 

5 

-3 

-r 

no  I 

— 

016 

— 

.013 

—  .  'JOI 

- 

.020 

— 

.052 

, 

6 

-3 

- 

(-03 

•    . 

- 

.002 

- 

.005 

- 

.(305 

2 

-4 

4- 

001 

.    . 

— 

.  002 

— 

.001 

— 

.001 

. 

—  I 

-4 

• 

+ 

4- 

OtI 

^  7- 

.017 
•174 

— 

.006 
.nil 

— 

.  006 
.02S 

• 

I 

-4 

- 

'X)I 

+        I 

7S5 

4- 

002 

—    1 

•495 

-«- 

.2f>I 

a- 

1. 177 

.-■' 

2 

-4 

— 

0,3 

^         2 

050 

+ 

025 

—    I 

■344 

Hi- 

■725 

4- 

3".7fi8 

,'- 

1 

3 

-4 

— 

:r,r 

•)-         2 

242 

— 

242 

4- 

.i/)5 

— 

IX) 

— 

2.7.J3 

-r- 

3S.426 

(■ 

4 

—  4 

— 

if>3 

4-         2. 

117 

- 

164 

4-     I 

■434 

- 

OfJ. 

- 

3.217 

-i_ 

1 3 .  IJOO 

w» 

5 

-4 

— 

030 

4- 

732 

— 

02y 

+ 

■533 

4- 

1.206 

4- 

1.9S1 

;■ 

■ 

—  4 
-4 

003 

4- 

"93 

OOf) 

003 

+ 

.oSG 
.oil 

■^ 

-173 

.020 

4- 
4- 

.221 
.023 

■■ 

TKANSFokM  \T„j.,^  O,.-  n.Wsr.NS  I.t'NAR  TUl.nKy. 


89 


«*icx 


4  (J  —  4  (,( 


—  I 

-5 

0 

-5 

I 

-5 

2 

-5 

3 

5 

4 

-5 

5 

-5 

6 

-5 

7 

-5 

: 

--f) 

2 

~6 

3 

-6 

-1 

-f) 

5 

-6 

1 

-7 

3 

-7 

-) 

-7 

4w- 

2 1'l' 

3 

0 

4 

0 

n 

—  1 

3 

—  I 

■» 

—  I 

5 

—  I 

6 

—  I 

—  I 

—  2 

0 

1 

3  -2 

4  -2 
1^     — 2 

6  -2 

7  -2 

8  -2 

1  -3 

2  -3 
-3 
--3 
-3 
-3 
—  3 
-4 
-4 
-4 


.rS„  ^ 


(«  ''  :)■'  X 


(''<':)-'X 


-t- 


+ 


!  ij  —  4  (J 

0  -3 

1  -3      + 

2  -3      + 


.002 
.011 

.UOI) 


— 

.001 

- 

.Ot2 

— 

.012 

+ 

.001 

+ 

•  o5<J 

i- 

.01 1 

-t- 

.002 

+ 


.001 
.oil) 
.oiS 
.003 


.IKJI 
OI)I 


.  0()() 
.004 


+ 
+ 

+ 

+ 

+ 

^ 


+ 


.001 

.012     — 
.011     — 

.  0(  )2 


.  I  >0 1 

.or  4 

•  I '13 
.2;ts 

•  230 
•244 
■"<)>) 
.013 

■.()(ii 
.010 
.UI7 
.016 
.ni7 
.(JoS 
.001 

.  DO  I 
.001 


,  00 1 

.003 
.063 
.032 
.0(10 

■"37 
.002 


.  (X13 
.003 
.  003 


+  .005 

+  .OfJI 

—  .011 

+  .052 

+  •"55 

+  .011 

+  .001 


+    -578 
-8.f)64 

-5 -7-14 

—  1 .1)01 

—  .126 

—  .on 

—  .001 

+    .027 

—  -377  , 

—  .374 

—  .070 

—  .001) 

—  .001 
+    .1)1)1 

—  .013 

—  .016 


.001 
.  00 1 


.f.S, 


—  .01  t 

—  .007 

—  .001 


-t-.oi3 
—  .003 


I'-.V): 


+  .002 
—  .021 
-.(,>I7 
- .  003 


— 

.fXJ2 

.017 

- 

.141 

— 

.lf)I 

+ 

.o()S 

■V 

.106 

+ 

•f>73 

-t- 

.012 

+ 

.  00 1 

— 

.001 J 

- 

.1)12 

(- 

.  1  )03 

•+- 

.012 

+ 

.006 
.001 

+ 

H- 


f 


—  .001 
+  .011  — 
+  .007  — 
+  .001 


—  .008 

—  .001 


—  .002 

—  .002 

■  !  I 

+       .  002        —  .  (X)2 

—  .ooS      +      ,001      —.002 

—  .004      —     ,003 


Sum. 


'<'.■•>         ('•,A').n 


003 

(HJ3 

004 
'W)3 


-t- 
+ 
+ 

+ 
+ 
+ 
+ 

+ 


•    • 

+  .004 

■     • 

+  .032    • 

+  .025 

•     • 

+  .0(J7 

+  .002 

•  003 

■"53 

.026 

+  .001 

.050 

-r  .  004 

.029 

+  .  004 

.003 

+  .001 

.  002 

.001 

—  .  002 

.002 

—  .030 

.002 

—  .025 

+ 
+ 
+ 


.001 

.003 

.025 
.080 
.262 

■  378 

.lf)S 

.025 

.002 
.out 
.005 
.017 
.02S 
.014 

.0(X) 
.(K)I 
.002 


TcMjis  in        .i 

Kcli|)iii  l.iiii-     •=" 

giiudu.  I 


.o<JI 

.003 

.072 

2.746 

4.4"2 
I  .SS6 
.286 
.031 
.002 
.  on  I 
.156 

•  3n 
■  153 
.024 

.(XX) 

.017 


+ 

+ 

-t- 
-t- 
-I- 


+ 


+ 
■(- 

■+- 
+ 
-t- 

+ 


.OC5 

.(XJI 

.<X)4 
.  062 
.064 
.015 
.003 
,001 

.OOl 

.009 

■  55S 
8.76(1 
5-756 

•  964 
■124 
.014 
.001 
.001 
.021 
■3S1J 

■  3''6 

■  "75 
.010 
.001 
.001 
.015 
.017 


.001  I  — 
.002  — 
.003   — 


n 


•  005 
.0(JI 

.002 
.070 
.064 
.015 
.  003 
.001 
.001 
.  (.«)6 
■534 
9  37" 
5-743 
■V91 

-124 

.014 
.001 
.001 
.023 
•430 
■3S4 

.075  j 
.010  I 

.OOl 

.001  ' 
.015  ■ 
.017  j 

.001 


■  025 
.015 


"33- 


90 


rU.WSIOUM A  1  K  IN  or   IIWM'NS  I,r.\\K   rilKOKV. 

Taiilk   II. —  Tin    Milan's   /,n)ii/i/iii/f' — ( 'oiitiiiiu'd. 


//  .1 :  X 


A'     A' 


.1  — 

4i„ 

-  1 

(1 

_4 

1 

-4 

3 

-  1 

3 

-4 

4 

-< 

5 

-4 

I 

-5 

2 

-5 

3 

-5 

I 

-f) 

/S„ 


.(104 
•  I '3  3 
•275 
•M3 
.01 1 
.  001 


.001 
M2r 
.UI6 
.  I  II  I  I 
.001 


('■.  C)i 


.1113 
.(II 1 

,(H)■^ 
.014 

■  ii"3 


."'11 

.  on  t 
.n()2 


(//ii;)-X  ('/''-./'X 


<'•..<'):  Ui..'  (-..C):! 


.0 
'Irl  Ml-,  ill  _    ti 

r.i  li|itli'  1.1)11-     .=-'0 
Killlil''.  ^    u 


.(Mil 

,            , 

,      , 

4- .<MI| 

.       .         4- 

.004 

"14 

.002 

- 

.010 

1  .02*1 

.       .        — 

.  (  i(  1  4 

1172 

t 

,  (  M  )  J 

„. 

."-3 

1   .  12(1 

. 

.022 

"■' 

( 

.  idi, 

- 

.111; 

l-.(ii/l 

1 

.(i(i() 

i.lS 

(• 

•133 

+ 

■"?3 

>  .('2.1 

f- 

.(.S'l 

oor 

f- 

.011 

■t 

.(.24 

. 

-t- 

.1137 

OU2 

. 

,        . 

— 

.(K)fl 

(Hl2 

•      . 

.      . 

. 

.002 

llnl 

. 

■)~  .<M2 

4-  . 

.0111 

l"'7 

- 

.no; 

4-. Oil 

— 

.001 

ooS 

j.- 

1123 

- 

.  ( l<  '2 

4-. (lis 

.        4- 

.(JO(l 

(ic,3 

+ 

'"5 

i 

.00; 

t-.oo.-) 

4- 

.005 

(- 

.(«)3 

. 

•        -1- 

.  1  '(14 

• 

4- 

001 

•     • 

. 

.  uoo 

+ 


.003 
.  o(  1 4 

. '  M)2 
.22.1 
,  0(  1 1 
.o2() 
.(i|  I 
.(i(i2 
.  ( H )  I 

.  ( 10 1 
.027 

■<J35 
.1K14 
.000 


3 

~~  5 

4 

""  5 

5 

~~5 

3 

-6 

4 

-6 

S 

—  () 

.  IJI 13 


.  00  3 
.001 
..10.4 
.  ( 1' )  I 

•   ' 

( )(  H  ) 

.  0(]0 

004 

.  n>  ;          , 

Ol)(> 

.oil 

(ID  I 

-t- 

.  006 

- 

.  *)(><! 

- 

.  I  >(  )  I 

.■ 

-(- 

.Oi)2 

.4 

'~5 

4 

5 

5 

~5 

I 

-fi 

2 

-fi 

.1 

~(, 

4 

-6 

5 

-6 

/ 

-f, 

2 

-7 

3 

-7 

4 

~  7 

5 

/ 

fi 

—  7 

7 

-7 

()  u  - 

-  4'.i' 

1 

-.1 

£ 

-3 

3 

^  1 

4 

-4 

5 

-4 

6 

-  1 

7 

-4 

S 

-4 

4- 

.002 

4- 

.oiS 

4- 

.0.41 

004 

4- 

■  "34 

005 

4- 

■"3? 

(iu2 

4- 

.  (-12  1 

•h 

.  o'  i6 

4- 

.1102 

4- 

.00; 

4- 

.006 

+• 

.  (;(j6 

-f. 

.004 

4" 

.OUI 

004 
004 
'Jl)2 


4-  .""I 

4-  .001 

—  .  01 1 1 

—  .<«>) 

—  .052 
r  .  0( )  I 
t-  .02S 

+  .(12(1 

+  .007 

—  .  ( 103 

—  .  0( )() 

—  .(K)3 

—  .001 


4- 

.Of)  I 

'   .002 

4- 

.001 

4. 002 

4- 

.002 

4 

.001 

4-  .004 

— 

.05^1 

- 

001 

-•"1)3 

— 

.o>o 

— 

.001 

-.Uf) 

— 

•  "34 

-.052 

~ 

.""7 

—  .012 

-  .(Kl2 

4- .  002 
+  .ooC) 
4  .oo.S 
4-  .oo3 
+  .ooij 
4-  .o(i; 

^  .001 
f  .001 

r  .  00  J 
4-  .O'JI 
4-  .()OI 


4- 

.001 

- 

.003 

-u 

.001 

- 

.nio 

— 

.001 

- 

.(10() 

.000 

.000 

+- 

.  ()02 

+ 

.011 

4- 

."07 

4- 

.2(J2 

4- 

."30 

+ 

•572 

■-t- 

.0(13 

+ 

•3()5 

(- 

.042 

4- 

.126 

4- 

.(JI4 

4- 

.023 

.000 

.  (J(X) 

.000 

-r 

•"37 

4- 

.o()| 

+ 

•  oSq 

4- 

.  oof) 

+ 

.067 

4- 

.004 

4- 

.020 

4- 

.001 

4- 

.001 

4- 

.003 

4 

.(J03   !      . 

-)- 

.003 

4- 

•""3  1  ^  . 

4- 

.  oof) 

— 

.005  i     , 

— 

.  ICO 

— 

•  lf)6  \     . 

- 

.  KjS 

- 

■  203  I     . 

— 

.o8f) 

- 

.086  '     . 

— 

.OK) 

- 

."19 

— 

.002 

— 

.002 

•n<.\NSFu,n,\,,oNor,lXNSKN'Sl.rN.M,  Ti.Kouv. 


t;l 


.)  u 


3  I) 
•»  <) 
?  I' 
^  ., 

4  I 

5  I 

■t  u' 

"  A 

4 

(l  u  — 2  (./ 

4  -2 

5  -2 

6  -2 

7  -2 

S  (,)  — 6  (./ 

5  -f' 

6  -f, 

7  -0 

5  "-3  "' 

3  -3 

4  -3 

5  -3 
(>       -3 


■iS. 


(»/rls).' 


('■..Or- 1  \i 


i)  I  J  S| 


('■..V): 


<>  1.)  — 4  u' 

4       -5 

•      ■ 

5       -5  ■ 

.      . 

('       -5 

. 

7       -5 

•      • 

4  (.)  —  ')  r,i' 

I         -(, 

. 

2          ~(,        - 

.(X)3 

3       -f.      - 

,004 

4       -6      - 

.CJl)2 

002 
(XJ4 


.005 


—  .i«'i      -t. 111)5 

—  .001       (  .(103 

-H  .  I102 


—  .1121 
+  .  I („) 
+    .  020 


.002 


+    .002 
+    .1)112 


■t-     .0O2 
+     .007 


—  .fll)| 


—  .(XKJ 

—  .ix)7 

—  .  (X)3 


""3 

002 


—  .002 
— .  003 


+  .003 
+  .002 


—  .002 

—  .  002 

—  .002 


(H.l-.)' 


('■,;■). 


Siiti) 


1. 

liriiis  III 

l.iliplir  l.iiii. 

Kitudv. 

'•J  ^ 
~  g 

P 

1)14 

.014 

022 

—            .022 

, 

..tj 

—          .oia 

. 

002 

~          .1x13 

4-  .  002 

.         + 

.0O2 

t- 

.001 

f    ,002 

.         + 

•  o<J3 

+ 

.Ol)^ 

•f.002 

f- 

.(M)2 
.IHHJ 

+ 

.(MJ5 
.  00 1 

+ 

.I>02 

.<«K) 

- 

.017 

- 

.010 

+ 

■174 

+ 

.oS2 

-f 

.022 

+ 

.423 

•t- 

.Ci,2 

^■ 

■'>')4 

.       . 

+ 

•  013 

— 

.002 

.  ( H  HJ 

. 

+ 

.(KJl 

.004       — 

•  0113      - 


.005 

.CH)4 


+ 

4- 


.(X)4 
.o<j3 


.0<J2 

.020 
.013 
.002 


(KJ2 

— 

.O-li 

(X)3 

- 

.004 

no2 

- 

.003 

1 

.     + 

.002 

+ 

.cxja 

-f-  .  OOfi 

.     + 

.013 

+ 

.013 

+  .004 

.    + 

.004 

-t- 

.004 

+  .001 

.     + 

.001 

+ 

.<X)I 

TRANSroKMAIloN  ol'  IIANSF.S'S  1,1  NAK  TIII.OKV. 


TaULI;  111. —  Hiihind  ( 'iirj/itifiity  11/  Li)Hifihiili;  tinit)<linij  ti>  IIaNsKN  (llld  HKIiAWNAY 

!    ■ 


S  S 


I/iinsfii. 


II). 


(a). 


1);-    I),  II  -   I). 


I 

0 

asian.rs 

22640,15 

22640.15 

. 

3 

0 

■(- 

yfxj.o') 

4- 

769.12 

4- 

769.()() 

- 

() 

0 

3 

1) 

+ 

3f>i3 

4- 

36.16 

4- 

3f).ia 

- 

4 

4- 

I 

4 

1) 

-f 

i.()4 

4- 

1.96 

( 

1.94 

- 

a 

0 

5 

0 

1- 

1).  11 

4- 

0,  13 

4- 

0.  II 

- 

I 

0 

fi 

0 

+ 

o.ol 

4- 

0.01 

+ 

O.OI 

0 

0 

-.1 

—  1 

+ 

(i,i>4 

f- 

o.o( 

, 

, 

—  3 

-1 

+■ 

0.55 

t- 

0.52 

4- 

0. 56 

4- 

4 

- 

—  2 

—  I 

+ 

7.f'7 

4- 

7.f'2 

•t- 

7.69 

4- 

7 

- 

-I 

-I 

4- 

1l)().')2 

4- 

10  J.  79 

4- 

lOfi.Sj 

4- 

u 

4- 

0 

-I 

+ 

fi6().s9 

4- 

66c,.  57 

4- 

f69,76 

4- 

") 

4- 

I 

- 1 

4- 

MS. 1)2 

4- 

Ul\(> 

I 

14^  43 

4- 

97 

- 

41 

2 

-I 

4- 

(,.72 

-i. 

'>.5') 

1 

9.7" 

4- 

13 

+ 

3 

-  I 

4- 

o.^>7 

4- 

0,63 

4- 

n.66 

4- 

3 

4- 

-1 

-I 

4- 

0.05 

4- 

0.04 

,       . 

. 

0 

n 

4- 

(J,  1)6 

4- 

0.07 

.       . 

. 

, 

—  I 

—  2 

4- 

I.I> 

-f- 

1.16 

4- 

I.  16 

0 

4- 

0 

—  2 

■1- 

7.M 

+ 

7.4<) 

4- 

7.46 

- 

3 

+ 

I 

—  2 

4- 

2.?l) 

4- 

2.49 

4- 

2.59 

+ 

10 

2 

—  2 

4- 

O.I.> 

-f- 

0.  16 

. 

. 

3 

—  2 

+ 

0.01 

4- 

0.01 

, 

. 

~i 

-3 

4- 

0.02 

4- 

O.OS 

,       , 

, 

• 

0 

-3 

4- 

o.oS 

4- 

0.14 

,       , 

• 

I 

—  3 

+ 

0.05 

4- 

0.03 

•       . 

. 

• 

2(.i  — 

21.1' 

—  I 

0 

— 

0.01 

— 

0 . 0 1 

,             , 

. 

. 

0 

0 

- 

0,2'-, 

— 

n.  16 

1 

0 

— 

2.54 

- 

2  .  22 

- 

2-35 

4- 

•3 

■>; 

2 

0 

— 

o.fi) 

- 

o.)5 

— 

0.15 

0 

■f 

3 

0 

— 

0.01 

- 

o.or 

.            , 

, 

„  '^ 

-1 

4- 

0.02 

. 

—  I 

—  I 

-f 

0.18 

+ 

0.07 

,           , 

. 

<t 

-  I 

4- 

2.?2 

4- 

..S7 

4- 

3.27 

4- 

40 

-r 

25 

I 

—  I 

— 

2S.5'') 

- 

29.50 

- 

28.32 

-I 

.18 

4- 

24 

2 

—  1. 

— 

24.4s 

~ 

24,60 

— 

24  ■  5" 

- 

10 

— 

3 

-I 

— 

2.')3 

- 

2.9') 

- 

2.96 

0 

— 

4 

-I 

— 

0.21, 

— 

0.27 

, 

, 

5 

—  1 

— 

l).02 

- 

0.02 

,     , 

, 

-3 

—  0 

4- 

0.07 

4 

0 .  f  )6 

,     , 

_  2 

—  2 

4- 

o.(;5 

4- 

0.9I 

•t- 

1. 00 

4- 

') 

-. 

—  I 

—  2 

4- 

■3.1'J 

4- 

13.15 

4- 

13  32 

4- 

17 

- 

'3 

0 

—  2 

4- 

211.71 

4- 

2lt.46 

4- 

211.84 

4- 

38 

13 

I 

—  2 

^f 

45S6.e6 

+ 

45^6.24 

4- 

4SS6.44 

4- 

20 

1- 

13 

2 

—  2 

4- 

2369.75 

4- 

2369.71 

4- 

2369.74 

0 

4- 

3 

—  2 

4- 

191.95 

4- 

I92.(X) 

4- 

192.00 

0 

_ 

4 

—  2 

4- 

14.3a 

4- 

14.  0 

4- 

14.40 

0 

_ 

2 

S 

—  2 

4- 

I  .  of) 

4- 

1 .  06 

4- 

1 ,06 

0 

0    ; 

6 

—  2 

+ 

0.08 

4- 

O.oS 

•       • 

• 

• 

TRANSFORMATION  OF  HANSEN'S  MSAK  TJIFORV. 

'I'ai.u.:   hi,-/,',,/,,,,,,/  (%„.llinn,f.s  uf  Lo„i,H,ul,,  ,(v._C„„fiimccl. 


Iliinun. 


(I). 


8u  - 

•*"' 

,, 

• 

1    -a  ' 

-3 

+ 

0.03 

+ 

".03 

-I 

■""  3 

+ 

0.48 

+ 

"••»') 

+ 

0.49 

o 

+ 

S.fif) 

-f 

a.M 

+ 

a.fifi 

I 
a 

"~  3 

+ 

Sofi.4f) 

+ 

2of,,54 

+ 

206.34 

~3 

+ 

If)5.f.3 

4- 

165.55 

+ 

If'!.  55 

3 

~3 

+ 

I4.f)0 

+• 

^•5'l 

+ 

14.66 

1        •» 

"~*  3 

+ 

1.18 

-f- 

I. II 

•f 

'•i; 

S 

—  3 

+ 

0.  in 

+ 

n.o? 

1     -' 

—  4 

■t- 

0,02 

+ 

n.oi 

t        n 

1 

—  4 

+ 

0.28 

+ 

n.2S 

I     I 
1 

—  -4 

H- 

7.44 

+ 

"•5" 

+ 

"•  5" 

1       a 

-~i| 

+ 

S.13 

•+- 

S.nfi 

+ 

S.06 

3 

— 4 

+ 

0,76 

1      * 

0.68 

+ 

0.72 

A 

—  4 

-t- 

0.06 

+ 

C.05 

0 

—  5 

+■ 

0,01 

I 

—  5 

4- 

0.26 

-H 

n.  i() 

•     • 

2 

—  5 

+ 

"•34 

+ 

11,2? 

3 

—  5 

+ 

o.o^ 

!     + 

0.01 

, 

;        2  (J 

() 

+ 

n.ot 

+ 

n.o2 

I 

- 

0.0. » 

- 

0 .  09 

, 

i       a 

+ 

0.  4? 

+ 

n.42 

+ 

n.42 

i       3 

+ 

0.27 

+ 

0.26 

4 

+ 

0.04 

+ 

0.04 

—  t 

0 

+ 

0.07 

+ 

0,05 

+ 

0,05 

0 

0 

+ 

1 .09 

+ 

i.3'J 

+ 

■•33 

I 

0 

— 

3<>.5S 

- 

39.54 

- 

3'^.5t 

9 

0 

— 

411  .f)0 

— 

411.63 

- 

411.63 

3 

0 

— 

45-'"3 

— 

45. »2 

— 

45.12 

4 

0 

— 

4 .  no 

— 

4. or 

~ 

4.01 

5 

0 

— 

"•33 

- 

".33 

- 

".33 

6 

0 

— 

0.03 

— 

0.03 

. 

0 

—  I 

+ 

0.07 

— 

O.OI 

I 

—  I 

+ 

o.nS 

+ 

0.12 

2 

—  t 

— 

o.oS 

- 

0.09 

,      , 

3 

—  I 

— 

n.3,) 

— 

n.28 

.      , 

4 

-I 

— 

0.05 

— 

0.04 

, 

5 

—  I 

— 

O.OI 

,      , 

,     , 

^u' 

—  I 

4 

- 

n.ni 

+ 

n.oi 

0 

4 

— 

0.07 

— 

0.07 

I 

4 

+ 

u.or 

, 

—a 

3 

+ 

"■"3 

u. 

n.03 

. 

—I 

3 

+ 

n .  40 

+ 

"•37 

+ 

".37 

o 

3 

— 

2.15 

— 

2.17 

— 

2.17 

I 

3 

+ 

O.of) 

-i_ 

0.05 

3 

3    1 

i 

+ 

0.03 

+ 

0.02 

•      • 

I). -I),    II    I), 


+ 


o 

n 

20 


I 

o 
12 

3 
6 

3 


-       ft 

+       7 
+       4 


+ 


+ 


2 

29 

4 
3 
3 
I 
o 


93 


94 


rR.\NSF()KMATU)N  <)1-    IIANSKNS  UNAK  TIIKOUV. 

Tai'.i.k  III. — liiilnad  l't,f[liiii'}ils  of  L<i)i(iitHilc,  &.C. — CoiitiiuuMl. 


K 

A'' 

// 

DIU'll. 

Di- III  II  nay 
(1). 

Ih'/iiiiiiiiy 

12). 

0.- 

-n, 

11  - 

-  11 

,, 

tt 

„ 

2u' 

-3 

o 

+ 

0.03 

-r 

0.03 

. 

• 

-2 

2 

+ 

0.43 

-+- 

0.45 

+ 

0.45 

0 

— 

2 

-I 

2 

+ 

6.36 

+ 

f).37 

+ 

('■}- 

0 

— 

I 

0 

2 

— 

55  —  5 

— 

55.20 

— 

55.'7 

— 

3 

4- 

s 

I 

2 

_ 

o.iS 

- 

o.i8 

— 

0.14 

— 

4 

+ 

4 

2 

2 

+ 

0.56 

4- 

0.54 

+ 

".54 

0 

+ 

2 

3 

2 

+ 

0. 10 

.  + 

o.oS 

• 

• 

4 

2 

+ 

O.OI 

+ 

O.OI 

.      . 

. 

. 

-2 

— 

n.in 

— 

O.OI 

. 

. 

. 

-I 

— 

u,  1)8 

— 

0. 10 

. 

. 

0 

+ 

1 .  55 

■+- 

1.43 

+ 

"•43 

0 

+ 

12 

I 

-\- 

I). 1)1 

+ 

O.OI 

2 

+ 

O.OI 

, 

. 

. 

. 

0 

0 

+ 

O.OI 

+ 

0.02 

•      • 

• 

• 

2U  +  2 

u' 

I 

2 

- 

0 .  03 

. 

. 

. 

. 

2 

2 

+ 

0 .  oS 

+ 

O.oS 

.      . 

. 

. 

3 

2 

o- 

O.OI 

+• 

0.02 

•      •     ' 

• 

• 

(J  —  G, 

o 

-f- 

O.OI 

,      . 

. 

I 

- 

0.03 

- 

0 .  04 

.      . 

•' 

o,(X) 

— 

0.01 

-2 
-I 

+ 

+ 

0.02 
0.33 

+ 

0.02 
0.26 

•      • 

(.) 

-! 

1.33 

+ 

0.S7 

T 

0.S7 

0 

+ 

46 

1 

-:- 

18.09 

+ 

18.08 

+ 

18. oS 

0 

+ 

I 

o 

+ 

1.27 

+ 

1.22 

+ 

1. 21 

— 

I 

+ 

6 

3 

+ 

0.09 

+ 

0.09 

. 

4 

+ 

O.OI 

+ 

O.OI 

. 

-3 

—  I 

— 

O.OI 

, 

, 

. 

_'> 

-I 

- 

0.  12 

- 

0.09 

. 

. 

-I 

-t 

— 

I.  73 

— 

1.50 

— 

1.59 

-^ 

') 

+ 

10 

() 

-1 

- 

IS. 70 

— 

•S.35 

- 

IS   76 

~ 

.'I 

— 

6 

I 

-r 

- 

125-43 

■- 

125. 4<) 

- 

25. 98 

+ 

4'J 

— 

55 

2 

-I 

- 

S.43 

- 

8.43 

- 

S.54 

+ 

9 

- 

6 

3 

—  I 

— 

0.59 

— 

0.57 

— 

.60 

+ 

3 

— 

I 

4 

—  I 

- 

0.04 

— 

0.04 

—  I 

—  2 

— 

O.OI 

— 

O.OI 

, 

o 

2 

- 

0.17 

— 

0.14 

— 

0.14 

0 

+ 

3 

I 

—  2 

- 

o.Oo 

- 

U.55 

- 

0.56 

+ 

I 

+ 

4 

2 

—  2 

- 

0.13 

- 

O.oS 

, 

. 

, 

3 

—  2 

- 

0.02 

— 

O.OI 

,      , 

, 

, 

I 

-3 

-t- 

0.04 

+ 

0.05 

.      • 

. 

. 

3"-. 

1  u' 

+ 


4        -3 


o .  04  — 


U.02 


0.14 
0.03 


TR.\\SR)RMAT!..X  or  II.WSKNS  I.fNAR  TUKnny. 

Tm-.u:  III.—/',,/,,,.,,/  fV///V;r///.v  n/ Lu>H,;/m/r,  .(r.— CoutiiuuMl. 


95 


IIiuihh, 


V:' : 

o 
I 

3 
4 


-3 

-3 
-3 
-3 
-3 
—  3 
-4 
-4 
-4 
-4 
~5 
-5 


+  '.I 


3 
4 

2  

3  - 
4 

w  —  3  (,)' 

1  —2 

<>         -3 
-3 

2  -3 
1          ~4 

4  '.I  —  4  '■>' 


3 
I 

3 
4 
5 
-t 
o 
I 

3 
4 


-3 
-3 
-3 
-3 
-3 
-4 
-4 
~4 
-4 
-4 
-4 


+ 

+ 


(^' 

«)' 

o 

41 

o .  n8 

— 

0.23 

— 

(1 .  r)f) 

+ 

+ 

0.01 

-t 

0.32 

0.01 


u .  ( )3 
0.02 
0.01 
0.36 
n .  f  14 
'-),  29 
0.05 
0.0 1 
0.03 
I.IS 
30.73 
3S.43 

13.  (JO 


Delaiiiwv 

Pi-htiitiay 

(I). 

(2). 

0.04 

" 

1. 17 

— 

1.23 

2 .  (;S 

— 

3.12 

"'•57 

+ 

0.54 

0.04 

+ 

0.01 

O.OI 

0.07 

o.iS 

O.II 

O.Ol 

I), 


+ 

O.OI 

+ 

o.oS 

+ 

+ 

0-55 

4- 

-t- 

O.OI 

+- 

H 

".05 

0 

+ 

0.06 

0 

— 

0.04 

.. 

n 

— 

O.OI 

— 

I 

+ 

0.02 

+ 

r 

+ 

0.25 

-t- 

1 

+- 

0.04 

-t- 

4- 

+ 

+ 


0.04 
0.5,, 
0.03 


o .  04 

<l.O! 
<»,02 
0.24 
0.04 

O.OI 
0.03 
0.26 
O.OI 

0.02 


O.OI 
O.OI 

0.02 
o.()7 
0.S3 
o.  29 
0.04 


0.(/) 

30.52 

3^.31 

i3.S(j 


+ 


0.5., 


-f    I 
—     I 


0.67 

0 

—    31 

0.S3 

0 

-    1.) 

0.30 

+ 

1 

—       I 

+ 


t.oS 

+ 

12 

4- 

10 

30.72 

■f- 

20 

+ 

h 

3S.48 

+ 

17 

— 

c 

13.98 

+ 

9 

— 

8 

96 


•IKANSrOKMATION  Ol"  I  lANSKNS  LKNAR    TIIRORY. 

Taiile  lU.—li.ediiced  CovJJicicnh  of  Loii(/ititdr,  rCr.— Continued. 


s 

A'' 

lUlHS 

■«. 

Diliiiiihiy 

llctii'inav 

(2).    ' 

D:- 

I), 

11  - 

1 

0, 

4  ,j 

-4  "•     ! 

.. 

i, 

" 

1 

5 

-4 

+ 

i.yS 

+ 

I.S6 

4- 

I.  88 

+ 

2 

4- 

10 

(> 

-4 

+ 

0.22 

+ 

o.iS 

4- 

0.20 

4- 

2 

4- 

2 

7 

-4 

+ 

0.02 

-*- 

0.01 

. 

. 

I 
3 

-5 

—  5 
-5 

+ 
1- 

0.07 
2.75 
4.41 

4- 
+ 
+ 

o.oft 

2 .  fto 

4.28 

4- 
+ 

2. 75 
4-3! 

+ 
+ 

ft 
ft 

4- 

0 

7 

4 

—  5 

+ 

I.S9 

+ 

1.67 

V 

1.71 

4- 

4 

4- 

18 

5 

-5 

+ 

CJ .  2() 

4- 

0,20 

•       • 

6 

■~  5 

+ 

0.03 

4- 

O.OI 

2 

-6 

+ 

O.lft 

4- 

i ) .  1 1 

3 

-6 

-4- 

0.31 

4- 

0    22 

4 

-6 

+ 

0.15 

+ 

0.  10 

5 

-ft 

+ 

0.02 

+ 

O.OI 

1 

3 

-7 

+ 

0.02 

•      • 

•     1 

4 

/ 

+ 

O.OI 

•        • 

4  "' 

-2  i,l' 

O 

-I 

, 

+ 

O.OI 

.      . 

3 

—  1 

+ 

0.07 

-t- 

0.  II 

■       ■ 

4 

—  I 

+ 

0 .  oft 

+ 

0.07 

•      • 

5 

—  I 

+ 

0.02 

4- 

O.OI 

,•      •      i 

I 

—  2 

+ 

O.OI 

. 

2 

—  2 

- 

0.54 

— 

0.54 

— 

0-53 

— 

+ 

I 

3 

2 

- 

9-37 

- 

9-34 

— 

9-3'J 

4- 

- 

2 

4 

—  2 

— 

5-74 

— 

5-73 

— 

?-73 

0 

4- 

I 

5 

—  2 

- 

0-913 

- 

0.98 

— 

I  .  ( K) 

4- 

- 

I 

6 

—  2 

— 

0.12 

— 

0. 12 

•         • 

7 

1 

— 

O.OI 

— 

O.OI 

■ 
•        • 

. 

1 

-3 

- 

0.02 

- 

0.02 

■         ■ 

3 

-3 

- 

"■43 

- 

0-43 

— 

0.43 

0 

0 

4 

-3 

— 

0.3S 

— 

0.37 

. 

5 

-3 

o.oS 

— 

o.oft 

. 

f) 

-3 

- 

O.OI 

•         • 

. 

3 

-4 

- 

0 .  02 

— 

O.OI 

•         • 

. 

4 

-4 

— 

0,02 

- 

O.OI 

<                • 

■ 

2  1.1 

—  J  w' 

-3 

0.03 

0.02 

2 

-3 

— 

<- .  02 

, 

o 

-4 

. 

4- 

O.OI 

I 

-4 

+ 

0.22 

4- 

0.34 

+ 

0-34 

0 

- 

12 

3 

-4 

- 

O.OI 

— 

O.OI 

0 

— 

I 

3 

■    4 

- 

0.03 

- 

<1 .  oft 

. 

4 

-4 

— 

O.OI 

— 

O.OI 

, 

• 

I 

-5 

i- 

0.03 

-+- 

0.03 

, 

2 

-5 

-t- 

0 .  04 

+ 

O.OI 

4- 

O.OI 

0 

4- 

3 

5  '■) 

—  5   6)' 

I 

3 

-5 

~ 

o.oft 

- 

0.02 

. 

, 

4 

-5 

— 

O.OI 

4- 

0.0a 

,       . 

, 

, 

5 

-5 

+ 

O.OI 

-T- 

0.02 

,      , 

, 

. 

3 

-6 

*~ 

O.OI 

•     • 

■    • 

• 

, 

Tk.WM-OKMATION  of  1|.\ns|;N'S  [.CNAR  tiikorv, 
TaI?I,K    III.— /;,.,/,,,.,,,/  CorJ/irinits  nf  Loiuiilndr,  ,(t.— C.iitiiiiUMl. 


97 


(>  U  —  (l  1,1 

//ij It  St  it, 

\ 

11  -  1). 

! 

4         -5 

—               O.OI 

—      0.01 

5         -5 

—                O.OI 

i          •  •  ' 

2            -f, 

+                  O.iJI 

I                > 

3         -f. 

+                 0.2IJ 

■t-      0.20 

4         -:< 

t-                0.57 

+              i'-4o           +              0.51            +      II 

+       f) 

5         —6 

+                 0.40 

+                0.2'> 

0         -6 

+                 0.  13 

+        "-07                 .    . 

7         -6 

+                 0.02 

+                O.OI                                  .       . 

3         -7 

-'                  0.04 

+             '^01                           .      . 

»        -7 

+                 O.OI) 

+             0.03                           .      . 

5         -7 

+           0.07 

4-             0.02                           .      . 

6         -7 

+           0.02 

f)  <J  —  4  (j' 

3         -4 

—            001 

~             0-"l                           .      . 

1 

4         -4 

0.17 

-             "-14                           .      . 

" 

5         -4 

—              0.20 

~             'J-  'f>                           .      . 

(,         -4 

~                    0 ,  0() 

—                  O.of]                                     ,        . 

7         -4 

—           0,02 

~                   O.OI 

-t         -5 

—                  O.OI 

—                   O.OI 

5         -5 

—                0.()2 

—                  O.OI 

6         -5 

—                 0,01 

• 

4(,i 

2                0 

—                    O.OI 

3            o 

+            0 . 1  )S 

4-             o.oS 

4            0 

+            0.42 

4-             0.42           +             0.42                    0 

5            o 

t-                 O.OiJ 

4-                            0.0(y 

f)                0 

-(-                   O.OI 

4-             0.01                           .      , 

fl  (.)  —  2  (.)' 

I 

5         -2 

+           0.02    j 

4-             0.02                           .      . 

6           -2 

+                  O.OI 

+                   O.OI                                        ,        , 

5  .J  —  3  '■'' 

i 

4        -3 

+            0,01     1 

I 

4-             O.OI      '                      .      .      I 

1 

■ 

98 


TKANb^l-'URMAriiiN  i)i'  II\.\M:NS  LINAK    Ilil.OKV 


TaUI-K     1\'.— ■/'//(      M.iHill's     /.illl/lli/r. 


j;      sin  1  sill  (f+u) 


sill    ( 


/  —  sill  A 


(I). 


/Kldtiihiy 
<■-•). 


u 

„ 

, 

<> 

,. 

ti 

n 

o 

3 

—  . 

,002 

, 

— 

.0(I2 

- 

.002 

.        . 

1 

3 

- 

.  003 

- 

.003 

- 

.003 

- 

•  "('3 

2 

3 

— 

.U^JI 

- 

.(»>1 

— 

.001 

•     • 

.        . 

—  r 

1 

— 

.  OU I 

— 

.001 

— 

.out 

— 

.004 

. 

o 

2 

— 

.0()2 

— 

oifp 

— 

.  Io> 

- 

.  1  oS 

- 

.075 

- 

o.oS 

I 

2 

— 

•  317 

4- 

262 

- 

■"5: 

- 

•  055 

- 

.072 

— 

0.07 

2 

2 

— 

.064 

+- 

.  009 

- 

•*''5 

— 

.051; 

— 

.055 

— 

n .  06 

3 

2 

— 

.  oof) 

. 

— 

.006 

— 

.  Ul  16 

— 

.006 

— 

O.OI 

^  1 

I 

- 

.004 

- 

I)2l) 

— 

.024 

- 

.024 

- 

.024 

- 

0 .  02 

—  I 

I 

— 

.U7I 

— 

233 

— 

■3"4 

— 

.001 

- 

.  3"-- 

— 

.300 

— 

0 .  30 

0 

I 

— 

3.089 

- 

573 

— 

5 .  662 

— 

.(-05 

— 

5   f'''7 

- 

5. 37" 

— 

5 -SO 

I 

1 

— 

30.U67 

4- 

23 

57^ 

— 

'1.4S9 

— 

.onS 

- 

'^■w: 

- 

6.471 

- 

6.33 

2 

I 

— 

(l.f.lO 

+ 

I 

27') 

— 

5-33' 

— 

.00^ 

- 

;.33f' 

— 

5-254 

— 

5-25 

_-, 

I 

~ 

.720 

- 

050 

— 

.(.40 

— 

.640 

— 

.617 

— 

0.62 

\ 

I 

- 

.116S 

4- 

00; 

- 

.063 

- 

.  063 

- 

.056 

— 

0 ,  06 

5 

I 

— 

.004 

. 

- 

.004 

- 

.(104 

— 

.  004 

—  I 

u 

— 

006 

— 

.006 

,(ii)() 

— 

.00^1 

— 

O.OI 

—  s 

o 

- 

.(H2 

- 

(•So 

- 

.01)2 

.092 

- 

• '  "15 

- 

0.09 

—  2 

0 

— 

•  254 

- 

I 

325 

■- 

1.5,^2 

— 

.003 

— 

1.5S5 

— 

1 .  590 

— 

1.59 

—  I 

(> 

- 

''.'J33 

- 

24 

7?7 

- 

31.72.1 

— 

.049 

— 

3I.7I") 

— 

31 -7*5 

— 

31.79 

O 

o 

- 

1020.614 

4- 

21 

1)1  (J 

- 

.,(jS.(.)5 

— 

.991 

— 

999.(j.56 

- 

<)99-747 

- 

9'*'J.75 

I 

o 

-r 

■  "444   '07 

0 

T 

lS44).fi07 

~l- 

lS.(-4i 

-t- 

18463.248 

4- 

18461.26 

+ 

1.^461 .26 

2 

() 

-r 

loio  337 

- 

I 

210 

+- 

tuoi).  121 

+ 

1.0:2 

i 

1010.173 

4- 

1010.233 

4" 

loio. 19 

.3 

1) 

"f" 

(j  1 . 9 1 ; 

- 

055 

M.S60 

+■ 

.0(1 

4- 

61 .901 

4- 

6 1 . 990 

4- 

61.99 

4 

u 

-r 

3  ■'(^3 

— 

003 

4- 

3.9^0 

— 

.  00 1 

4- 

3  ■97'; 

4- 

4.013 

-h 

4.01 

5 

0 

-•- 

.2(,3 

- 

.  263 

r 

■  263 

4- 

.  272 

+ 

0.27 

6 

( I 

+ 

.019 

-(- 

.019 

, 

4- 

.019 

4- 

.019 

+ 

0 .  02 

/ 

M 

.(jOI 

-r 

.001 

4- 

,001 

•> 

—  I 

+ 

.004 

021 

-t- 

.02; 

4- 

.025 

4- 

.024 

+ 

0.02 

—  I 

—  1 

t- 

.065 

J- 

246 

h 

.311 

4- 

.001 

4- 

.312 

-1- 

.316 

4- 

0.32 

o 

—  1 

+ 

3 .  26f) 

4- 

I 

^53 

f- 

5-  i"J 

u 

.lK)fl 

-V- 

5.125 

4" 

5.014 

+ 

5.07 

I 

-: 

+ 

2)-<-V 

- 

24 

V'i 

r 

4.571 

+ 

.002 

+ 

4.S>o 

+ 

4. '(=5 

4- 

4 .  so 

2 

—  i 

i- 

8.151 

- 

I 

y)<' 

4- 

('■755 

— 

.oo3 

-t- 

6.763 

+ 

6.5,9 

4- 

6.62 

3 

-  1 

4- 

.tSo 

- 

0S3 

f- 

•  7')' 

r 

.  (  K  >  I 

4- 

.79*5 

4- 

•714 

4- 

0.74 

1 

—  1 

4- 

.oSi 

- 

005 

T 

.070 

4- 

.076 

+ 

.o6( 

-t- 

0 .  06 

5 

—  I 

r 

.<xi5 

. 

4- 

.oo> 

+ 

.005 

h 

.004 

. 

1) 

—  2 

-: 

.040 

+ 

or7 

-t- 

•  057 

+ 

.057 

4- 

.061 

+ 

0 .  06 

I 

—  2 

4- 

■  35  5 

- 

\\(. 

4- 

.Oh, 

-» 

.019 

4- 

.037 

4- 

0 .  04 

2 

-2 

^ 

.142 

- 

( i2'i 

-L 

.116 

4- 

.116 

4- 

• '  "M 

4- 

0. 10 

3 

-2 

4- 

."1  > 

■*- 

.oi5 

+ 

.olS 

4- 

.  1.1 1 1 

4- 

O.OI 

4 

-2 

4- 

.001 

4- 

.on 

4- 

.001 

. 

I 

~'  3 

-(- 

.  004 

, 

4- 

.001 

4- 

.004 

-t- 

.002 

2 

-3 

4- 

.003 

4- 

.003 

+ 

.U03 

. 

002 

015 

1 '  1 

i- 

'>>- 

..03 

- 

12S 

— 

005 

.Of 

)2 

'4 

.  I. 

1 

'5 

002 

. 

01^ 

— 

.007 

— 

O.OI 

Ul|« 

- 

.083 

- 

O.oS  i 

131 

— 

.122 

— 

0.12 

005 

- 

.006 

— 

O.UI    { 

TK  ANSlORMAiloN   ol'   II  ANSKN'S  MNAR  TIH.ORV, 
IaI'.I.K     1\. --'////■   MiHiii's   L'l/iliii/r (  '(illtilllH-d 


99 


A 

,i 

.f 

.1 

sill 

s:n(/i,„| 

y 

siii  .i 

J 

-    sill     :i 

//,/.v.,v,v. 

/), 

(1). 

/) 

/iiiiii.iy 
(2).     ' 

(,)  — 

2  (.J 

.. 

„ 

,, 

„ 

-3 

-I 

+ 

.U04 

4- 

.004 

4- 

.(104 

_ 

.(101 

~- 

~' 

-t- 

.()2f. 

f 

.1)26 

+ 

.026 



.01 1 

_ 

(Jill 

~' 

~' 

^ 

.151 

— 

.(,'12 

-I. 

.(iSf) 

-1- 

.()S<) 



.,173 

_ 

(-I.07 

o 

I 

— 

.11' 11 

h 

.164 

- 

•  7 '.17 

. 

— 

.7'J7 

_ 

I  .  1(15 



1  .00 

' 

■  ')t<i 

1 1 .  1*^3 

- 

12.123. 

-- 

.017 

— 

12. 140 

— 

12.17,) 



1  2  .  1  S 

2 

~' 

^ 

.041 

— 

.75,) 

— 

.  S30 

- 

.0112 

— 

■  ^32 

— 

.,S2() 

_ 

0.82 

3 

~ 

.003 

— 

.058 

— 

.061 

— 

.(/it 

_ 

.o6(.) 



o.of) 

-5 

— _2 

"1" 

,001 

+ 

.001 

4- 

.001 

-4 

. o 

-(- 

.013 

+ 

.  0 1 3 

't~ 

.013 

4- 

.007 

4- 

O.OI 

—  3 

" 

i- 

■153 

— 

.,H(J 

-h 

•  131 

4- 

•131 

j_ 

.  116 

4- 

0.  12 

- 

—  - 

-i- 

I  .'>22 

— 

•  .3"3 

+ 

1.51,1 

4- 

1. 51,) 

4- 

1.4;,) 

4- 

I.4() 

—  I 

—  - 

+ 

2  I  .  114  I 

-~ 

5  ■  4<J'> 

+ 

'^•55l 

4- 

.  01), ) 

4- 

I5.56(j 

4- 

I  5 . 362 

u- 

15.51 
166.57 

o 

2 

+ 

210.54,, 

— 

44.ufi2 

-r 

if.(i.473 

4- 

.12; 

-(- 

1 66 . 603 

4- 

l66.3,)5 

J_ 

1 

" 

+ 

9<).,jfio 

-f- 

5^=.5S4 

t- 

622.544 

4- 

t.lsS 

+ 

623.702 

+ 

623.(122 

4- 

623.62 

2 

2 

+ 

2.716 

.|_ 

3,1.  5SS 

~i- 

33.30i 

-1- 

.0115 

~ 

33  •3'' J 

4. 

33-3S2 

4- 

33  ^3? 

3 

—  2 

+ 

.  116 

+ 

2.027 

-f- 

2^I43 

+ 

.003 

4- 

2.  146 

4- 

2.  ifjo 

4- 

2.16 

4 

~- 

-1- 

.005 

-t- 

.14" 

4- 

.145 

•      . 

4- 

.145 

4- 

.14') 

4- 

(J.  15 

5 

~- 

-1- 

.(H.)l^ 

+ 

.1,(1 1 

4- 

.  (»,) 

4- 

.oil 

4- 

0.(JI 

-3 

-3 

-1- 

."05 

-f- 

.Of,- 

, 

4- 

.(1(15 

-H 

.003 

—  2 

-3 

+ 

.,/.3 

-- 

.007 

-(- 

."5^' 

+ 

.056 

,;^ 

.  0  i  2 

+ 

0.05 

—  I 

-3 

+ 

.S2f) 

— 

.171 

i- 

•  f'55 

+ 

■<>5  = 

+ 

•7I') 

-r 

().7() 

o 

-3 

1- 

<).2\<) 

— 

1-77^) 

+■ 

7-47'> 

4- 

.005 

-h 

7.475 

+ 

7.4''2 

+ 

7.  ^'> 

1 

-3 

1- 

(1.  ()IO 

f- 

22  773 

+ 

2i).6S3 

i- 

•n?3 

+ 

2,j.736 

4- 

2,).6S3 

4- 

2().6S 

2 

-3 

u 

.I,jS 

1- 

<-74 

+ 

1.772 

4- 

.(104 

+- 

1.77'- 

-L 

1  ,754 

4- 

'■75 

3 

-3 

i- 

. ,  „  )i  1 

+ 

.112 

-1- 

.  121 

i- 

.  121 

4- 

.124 

4- 

0,  12 

—  2 

-4 

t- 

.(,(J2 

+ 

.  0,J2 

. 

4- 

.  ( 11  >2 

+ 

.001 

-1 

-  i 

h 

.,,:', 

■      ■ 

+ 

.026 

4- 

.026 

+ 

.OKI 

4- 

0.02 

1) 

-  + 

t- 

•  331) 

- 

.,154 

-1- 

.276 

+ 

.  276 

4- 

.256 

^ 

0.26 

' 

-4 

..)- 

■337 

t- 

•75" 

+ 

I  .'i,j3 

f- 

.(),.'2 

4- 

1  .(11)5 

•r 

1  .(i>3 

4- 

1  .oS 

—  I 

-4 

-5 

.,11 1 

.Olll 

'■ 

.,,52 



+ 

.  "63 
.  (  ,1  1  1 

4- 

.065 

-r 

0.06 

O 

"~5 

+ 

(111 

4- 

.Ill  1 

. 

■f 

.01 1 

-f- 

.(-:(!-, 

■    , 

I 

~5 

+ 

.,)I4 

4- 

.1114 

t- 

.014 

+ 

.028 

4- 

0.03 

1.1  +• 

2i.i' 

I 

4 

- 

.(«,; 

- 

.iii>; 

— 

.(101 

— 

.  1 104 

_ 

.0,13 

. 

() 

3 

i-. 

.,,2,) 

— 

.  ,„  12 

(-- 

.027 

— 

.(■(12 

4- 

.,125 

4- 

.oil) 

4- 

0.02 

' 

3 

— 

.14^ 

■r 

.1121) 

— 

.116 

+ 

.02a 

— 

.(i5S 

— 

.,i()I 

— 

U .  ()i) 

2 

3 

- 

.021 

-• 

.001 

- 

,  ,122 

+ 

.om 

— 

,012 

— 

.012 

— 

( 1 . 0 1 

3 

3 

— 

.,)02 

- 

.<nl2 

4- 

.o,)2 

0 

— 

.0111 

, 

-  r 

2 

+ 

.1111; 

-r- 

.11,13 

4- 

.  uoS 

— 

.(102 

-h 

.oofi 

+ 

.011 

4- 

0.0 1 

o 

2 

•t- 

■  S'l'i 

"- 

.  IMi 

+ 

.320 

- 

•  "34 

-t- 

..>'< 

-L- 

.2)1 

-t- 

0.2(J 

1 

2 

— 

3-5115 

h 

.7l,S 

- 

2.797 

-U 

.6(13 

— 

2.1,j4 

— 

2.1,)0 

— 

2.1,) 

- 

2 

— 

.(,,11 

f- 

•  'M3 

- 

.55S 

+ 

.232 

- 

.32(1 

- 

•313 

- 

0.31 

3 

- 

— 

•  "''3 

■- 

.063 

+ 

.(141) 

— 

.023 

— 

.023 

— 

0.02 

4 

2 

— 

.  ( 1, 1') 

. 

- 

.  (-K.,6 

4- 

,(105 

— 

.  ( 'O  I 

— 

.0(  ,2 

,        , 

1 1 

I 

— 

.  ,-ji  I'j 

- 

.IJlllJ 

4- 

.(1,11 

— 

.('U^ 

— 

.1105 

— 

O.OI 

1 

1 

-U 

.085 

- 

.ikiS 

4- 

.077 

— 

.(112 

)- 

.06  = 

4- 

■054 

4- 

0.05 

- 

I 

+ 

.,)(>(i 

^»  * 

. 

4- 

,<.i()0 

- 

.(-1(12 

-t- 

.004 

4- 

.005 

. 

3 

I 

+ 

.OlJl 

-h 

.UOI 

. 

4- 

.  00 1 

, 

lOO 


1  RANSl-ORMATION  OF  T  W'SKN'S  M'NAK  TIIFOkY. 

Taum-;   I\'. —  Thv  M«nii's  Lntilndc — " -iitiiuK'd. 


-4 
-4 

-4 
-4 
—  4 


sin  I  sin  (f+i. 


.()()2 
.oio 
■"33 

.(H)4 
.OOl 

,  (  >(  t  I 
.OM3 


sin  A 


(X)4 
"54 


u()3 


A  —  sin  A 


A 
llaiiSi-tt. 


0()2 

— 

ord 

4- 

(K)2 

- 

02i) 

-t- 

(1()^ 

- 

o;o 

f 

DiS 

+ 

OOl 

+ 

001 

(X)I 

- 

003 

-T- 

(K)I 

- 

003 

+ 

002 

- 

.002 
.(loS 
.024 
.068 
) 

.  ( 10 1 
.002 
•  OOt 


/h-lllHlhiy 


PehiHiitiy 
(2). 


002 

002 

ofi  \       -t 


006        (- 


O.of) 


3 

4 


! 

—  1 

2 

-I 

3 

—  I 

5 

—  1 

001 

nil 

ous 


THIS        r 
2&S      - 
SiS       + 
2t^2 
021 
(_»|2 


(H)2 
003 


+ 


003 

137 
002 


+ 
+ 
+ 


001 

. 

+ 

.001 

.oil 

+ 

.()(>  \ 

- 

.007 

— 

.1x15 

+ 

.ooS 

-f 

.013 

+ 

-(- 

.  oud 

+ 

.006 

+ 

. 

-(- 

.001 

+ 

.001 

+ 

.ooS 

— 

.001 

+ 

.1-11)7 

+ 

131 

- 

.023 

-r 

.  loS 

-t- 

816 

+ 

1 .1110 

— 

2 

.  Sofi 

— 

252 

- 

6. 05  I 

- 

( 

■  3"3 

— 

02I 

- 

1  .  000 

— 

1 

.021 

- 

002 

- 

.117 

- 

.11  ) 

— 

- 

.012 

— 

.1)12 

— 

— 

.o.)l 

- 

.  ( )0 1 

002 

+ 

,  1  )02 

(K)f) 

- 

.004 

+ 

.002 

■r 

,003 

- 

.003 

0 

- 

.007 

- 

.n()7 

— 

— 

.(M-)l 

— 

.  00 1 

— 

.  006 
.013 
.007 

.  Ol)  I 
.1K)I 

•  "  33 

2.fiJ7 

6 .  297 

I  .oi.S 

.119 

.OI2 


.ooS 


0.01 
0.0 1 
0.01 


0.13 
2.70 
f).3o 
1 .02 
o.  12 

O.OI 


.006       — 


I 

0 

— 

.oi_)5 

— 

.005 

— 

.005 

— 

.001 

2 

0 

- 

.  \\U 

- 

.llf) 

- 

.Il(. 

— 

.OIJO 

— 

0.09 

3 

0 

— 

•  o>5 

— 

.015 

- 

015 

— 

.009 

— 

O.OI 

4 

0 

- 

.001 

— 

.  00 1 

— 

.001 

0 

—  I 

- 

.  002 

-t- 

003 

+ 

.001 

-)- 

.001 

— 

001 

I 

—  1 

4- 

.  1 11 

- 

052 

-+- 

.  of  )0 

4.- 

.  0()0 

4- 

.U21 

4- 

0.02 

2 

-I 

— 

■  :74 

+ 

256 

- 

l.3'S 

- 

003 

- 

'.  .y.\ 

- 

I..'^ci2 

— 

1  .50 

3 

—  I 

— 

S.430 

-f- 

150 

- 

>   'So 

4- 

003 

- 

\.-zTi 

- 

I.3S2 

- 

1. 33 

4 

—  1 

— 

.259 

^ 

o<7 

— 

.242 

4- 

OO' 

— 

.  240 

- 

.239 

— 

0.24 

5 

—  1 

— 

.034 

— 

.034 

— 

."34 

— 

.023 

— 

0.02 

f. 

—  [ 

— 

.003 

— 

.003 

— 

.003 

. 

—  I 

—  2 

4- 

.IX)  5 

+ 

029 

+ 

.034 

4- 

■  "34 

4- 

."25 

4- 

0.03 

0 

-2 

— 

.004 

+ 

273 

+ 

.269 

4- 

.  269 

4- 

.240 

4- 

0.25 

1 

2 

— 

1.85^ 

4- 

23') 

— 

1 .622 

- 

01  )i 

— 

1 .623 

— 

1  •  73'J 

— 

1.68 

2 

1 

+ 

!<)<>.  4 7<> 

- 

296 

H- 

1 99 . 1  So 

+ 

303 

4- 

"'^■4"3 

4- 

199.277 

4- 

199.42 

3 

—  2 

+ 

"7-753 

— 

()(/i 

+ 

117.657 

- 

39') 

4- 

"17.258 

+ 

117.1SS 

4- 

117. 19 

I 

2 

-(- 

15.207 

— 

"15 

+ 

I5.l<)2 

— 

"77 

4- 

15.115 

4- 

15. 105 

4- 

15,11 

5 

—  2 

+ 

1.531 

— 

002 

-f- 

1 .  529 

- 

010 

+ 

I.  519 

1- 

1 .  502 

4- 

1.50 

6 

-2 

4- 

.141 

+ 

.141 

- 

001 

4- 

,14" 

4- 

.132 

4- 

0.13 

7 

—  2 

-f- 

.012 

4- 

,012 

+ 

.012 

4- 

.008 

4- 

O.OI 

■- 

--...„  , 

■_ 

TRANSFORMATION  OF  HANSEN'S  LUNAR  TIIF.ORY. 

Taiuk   ]\, —  ■/•/„.  M„f„f',,  Ltitifuilc—ContuniMl 


lOI 


,C'      sill  I  .siii(  /+  (,j) 


1 

2 

3 
4 

5 
fi 

2 

3 
4 

5       - 


3 

4 
3  (.,  - 

2 

O 
I 
2 

3 
4 

—  2 

—  I 
O 
I 

2 

3 
4 

5 

—  I 
o 

2 

3 
4 
2 

3 


-3 
-3 
-3 
-3 
-3 
-3 
-3 
-4 
-4 
-4 
-4 
4 
-5 
-5 

—  5 

4  w' 

—  2 

-3 

-3 
-3 
-3 
-3 
-4 
-4 
-4 
-4 
-4 
-4 
-4 
-4 
-5 
-5 
-5 
-5 
-5 
-5 
-f> 
-6 


I 
2 

3 
4 

5 
6 

4 


-fit 


-6 

-0 
-6 
-6 
-fi 
-6 
-7 


+ 

+ 
+ 


+ 

+ 


+ 
+ 
+ 
+ 


+ 

+ 
+ 
+ 


+ 
+ 


.071 

,,.iS4 

S .  1 80 

I.ldf) 

.  12S 

.1113 

.003 

•  334 
.4111 
.  oh  I 
.007 
.t  12 
.017 
.002 


.001 
.l)(U 
.003 
.010 
.005 
,t)l)l 
.  03 
.012 

•  "43 
.220 

•  f'"3 
.23(1 
.021) 

.(X)I 

.  002 
.U05 
.oiS 
.of/, 
.031 

•  <x)3 
.CKJ4 
.002 

.001 

.002 
.ix>5 
.005 
.001 


>1U  , 

> 

(  -  s 

in  ,i 

Ill'tSCIl. 

Deliiiititty 

De/ 

11111,1V 

(!)• 

(2).    ' 

^-          .010 

^ 

.010 

+ 

.010 

:    4- 

1* 
.010 

4- 

O.OI 

h              .oir> 

— 

.061 

. 

- 

.061 

'    .„ 

.07S 

(J .  oS 

-                              1  (    £ 

-        *.v>s 

-t- 

.014 

+ 

8.f;i2 

4- 

8.96S 

4- 

(J.  00 

.  1  r^  * 

.022 

-t-             " 

.012 

— 

.015 

\  + 

7  •007 

1    + 

7.<)4f' 

-1- 

7-95 

"^                 ' 

•«44 

— 

.(XJ4 

'<  + 

1 .  140 

4- 

1.0S2 

4- 

1.08 

• 

~ 

-  £3^ 

— 

.oot 

■  + 

.127 

4- 

.100 

4- 

0. 1 1 ) 

"^ 

.013 

1  + 

.013 

4- 

.  ou6 

+ 

0.01 

~ 

.003 

— 

.003 

1 

.  003 

."I4 

"^ 

.J20 

■  + 

.320 

1 

1   4- 

•  311 

4- 

0.31 

-Ul  1 

~ 

■3Tf> 

+ 

•3i)'> 

4- 

.3(^2 

4- 

0.36 

• 

"*" 

.fjfti 

+ 

.Of.l 

4- 

•043 

4- 

0.04 

• 

~ 

.'jr>- 

+- 

.  <  107 

4- 

.002 

— 

.012 

-f 

.012 

4- 

.005 

4- 

0.01 

'~~ 

.017 

+ 

.017 

4- 

.006 

4- 

0.01 

■ 

.f'OZ 

■+• 

.002 

.   . 

- 

-CjI 

_ 

.001 

.002 

'- 

.Q»>I 

+ 

.oot 

.  U02 

— 

.001 

.001 

_ 

.014 

„. 

0.0: 

•'4? 

— 

■»:3 

— 

•  153 

— 

•i'» 

— 

0.20 

.  OI(? 

— 

.  l.jj 

— 

.103 

— 

.114 



0.  II 

.UI2 

— 

- 

- 

.013 
•  003 

— 

.014 

— 

O.OI 

•      • 

— 

.012 

.OI2 

— 

.  002 

.045 

— 

.<j02 

+ 

.002 

_ 

.005 

.404 

-*- 

.624 

+ 

001 

+ 

•  625 

4- 

•  5S2 

+ 

O.fjl 

5  •957 

-f-                 f: 

.ffto 

H- 

016 

4- 

f..57f' 

+ 

^'•532 

t- 

(). ;() 

3^4?i 

+          3 

.t.*7 

— 

008 

+ 

3-f'70 

4- 

3^f'5> 

4- 

3.f'5 

•  44? 

4- 

.46a 

— 

002 

+ 

.4^) 

4- 

•  4'.4 

4- 

0.46 

■  047 

T- 

.045 

.'02 

mi 

+ 

.048 
.002 
.005 

4- 

.044 

4- 

0.04 

.01 1 

-i- 

.ryiff 

+ 

.02Q 

4- 

.042 

4- 

0.04 

•  45<-' 

^- 

.•If, 

-i_ 

0O£ 

+ 

•  517 

+ 

•  576 

4- 

0 .  60 

•  379 

-^ 

.41c 

+ 

.410 

4- 

•SOS 

4- 

0.40 

.050 

+ 

.053 

^ 

+- 

•053 

4- 

.046 

4- 

0.05 

.021 

+ 

.02; 

1 

• 

+ 

.025 

4- 

.02S 

+ 

0.03 

.01. J 

021 
OOI 

. 

+ 

.021 
.001 

4- 

.022 

4- 

0.02 

.CK>5 

~ 

oyj 

1 

+ 

.003 

4- 

.003 

.oOS 

-t- 

075 

+ 

•073 

+ 

.  t)6 1 

4- 

0.06 

.0H6 

+ 

091 

+ 

.091 

4- 

.  07 1 

4 

0.07 

.03; 

-t- 

036 

4- 

.036 

4- 

.(124 

4- 

0.(j2 

.002 

4- 

txja 

4- 

.  002 

+ 

.0112 

001 

4- 

.UOI 

4- 

.005 

lo: 


TKANSroRMATloN  tU'  IIANSKN'S  l.LNAK   l\il.'Hi\. 

'\\\u].v.  IN'. — The  M(i(ii/'-<   l.dlihcl' — ( "uiitiiiia-«l. 


^        ^'      sin  I  sill  (_/•+•''') 


sin  i3 


/;  -  sin  ,}         //,,„if„. 


•  I) 


/J 

/K/iiiiiiiiy 

(2). 


-3 

() 

-f- 

__  o 

O 

+ 

—  1 

IJ 

+ 

II 

4- 

o 

-t- 

—  4 

—  1 

— 

-3 

—  1 

- 

-■:!■ 

-  t 

— 

—  I 

—  1 

— 

—  I 

- 

—  1 

- 

-I 

- 

-2 

o 

*r 

—  2 

-3 


2<j  - 

"  (J 

2 

I 

O 

o 

1 

o 

o 

o 

3 

0 

4 

o 

n 

-' 

—  I 

-> 

- 

4 

—  1 

5 

—  I 

I 

—  2 

2  u  —  3  I.) 
<J  —2 


-3      - 


3       -3      - 


-4 
-4 
-4 


.Oo2 

+ 

.002 

.022 

, 

+ 

.022 

.Ol/j 

- 

.o?o 

•1 

.016 

•  777 

-h 

.015 

^■ 

.792 

.OOl^ 

+ 

.004 

1- 

.013 

.  ( M  J I 

— 

,  0(J  1 

.in; 

, 

- 

.015 

.151 

f 

.041 

- 

.110 

1. 174 

+ 

■  751 

— 

.423 

5  •  5"4 

+ 

,821 

- 

4.f,S3 

.083 

- 

.  500 

- 

.5S3 

.004 

— 

.030 

— 

."34 

.  i.j<  1 1 

. 

-t- 

.001 

,     . 

\- 

,oig 

+ 

.oil) 

.ot? 

— 

.00? 

— 

.020 

.002 

— 

.ooS 

- 

.oil) 

.002 

~t- 

.  002 

.  002 



.002 

.013 

+ 

.004 

-+- 

.017 

•  oiS 

- 

.  0()lj 

— 

.051 

.7'/' 

- 

.OOIJ 

+ 

.7S7 

.  lOI 

-t- 

.  101 

.010 

4- 

.010 

.023 

- 

.043 

- 

..)t.ri 

.4^2 

r 

.577 

-t- 

•  '25 

5.431 

+ 

.1S6 

— 

5-245 

•  Oj.) 

T 

.  UOIJ 

— 

.650 

...-(.3 

— 

.  of)3 

.  ot  )6 

- 

.  oof) 

.014 

-r 

.ooy 

- 

.005 

.03-^ 

+ 

.02() 

— 

.012 

.011 

. 

— 

.011 

.002 

. 

- 

.  002 

.IJ<J2 

-f 

.002 

.W'12 

— 

.  002 

.0'J3 

+ 

.  003 

.(J(JI 

+ 

.  020 

-V 

.021 

.  <-«)4 

, 

— 

.004 

.046 

- 

.002 

- 

.04S 

.*'7 1 

- 

.224 

- 

.295 

.W41 

- 

•  391 

- 

•  35" 

.IX  It 

- 

.045 

- 

.044 

.U03 

. 

— 

.003 

.  006 

. 

— 

.  oof) 

.006 

•    • 

+ 

.006 

+, 

+ 


005       — 
001      — 


OI>( 


+ 

4- 


+ 
)      — 


.00: 
.0x2 
.016 

•T'/3 
.013 
.001 

.  I  !'■ 

.423 

4.<::-- 
■  -'i 
-t'3l 

.tlOI 

.019 

.<Jifi 
.010 

.002 


.017 
.051 

.101 


+ 

+ 


.121       — 


.2S|        — 


.<.;o 
.1/03 

.00; 

.012 
.011 

.Orj2 


.<J2t 

.004 

-*95 
-3r' 
.044 

.003 

.006 


.UOI 

.«j|o 

+ 

.024 

)- 

.704 

4- 

.03; 

4- 

.Wj 

— 

,o^o 

— 

•3'« 

— 

4.75^' 

- 

.f)i.3 

— 

.039 

- 

.€02 

.010 

4- 

.oog 

— 

.018 

— 

.rxi2 

.014 

4- 

.043 

- 

•  7')' 

4- 

.101 

-u 

.ooS 

-f- 

.•/)2 

- 

.141 

4- 

3-323 

- 

-f>47 

— 

.o;3 

- 

.0.34 

..-13 

— 

..-x)4 

.ni2 

.033 
.002 
.034 
.2<yi 

■33') 
.031 

.0<JI 

.0(7 
.021 


4- 


0 

01 

0 

02 

n 

7') 

0 

03 

0 

01 

0 

oS 

1) 

3^ 

4 

S3 

0 

(jo 

0 

04 

0 

01 

0 

01 

0 

02 

O.OI 

0.04 
0.7.) 

o.  10 

O.OI 

o .  of) 
o.  14 

5.40 
o.f.7 

O.of) 


O.OI 
0!03 

0.03 

o .  2q 

o-3t 
0.03 

0.02 
0.02 


IK.WSFOKMATION  of.'  ||  \\sKNS  l.i;\.\K  TIIKoRV. 
IaIII.K     I\. 'I'lir     Mdiiu's     /ui/ihn/r ('( tilt  111  llcil. 


sill  I  sill  (  /H-(,.( 


sin  !i 


i1  —  sill  3 


//„ 


in. 


?'■'  - 

-  4 .,)' 

,, 

3 

-2 

- 

.002 

4 

—  2 

— 

.  01 )  I 

2 

-3 

-f- 

.005 

3 

-3 

- 

.030 

4 

-3 

- 

•'^55 

5 

-1 

— 

.«27 

C 

-3 

- 

.1.105 

I 

-4 

— 

.003 

2 

-4 

-h 

.OK) 

3 

-4 

+• 

2.ii; 

4 

-4 

+ 

3-"i7 

5 

-4 

+ 

1.201 

6 

—  4 

4- 

.2l(j 

7 

—  4 

+ 

.o2r) 

S 

-4 

+ 

.003 

3 

-5 

+ 

.21S 

4 

-5 

+ 

•347 

5 

5 

+ 

.  1 62 

Ci 

-5 

+ 

.031 

7 

—  5 

+ 

.003 

3 

-6 

+ 

.012 

1 

-6 

+ 

.024 

5 

-0 

+ 

.013 

6 

-fj 

+ 

.002 

4 

'~  1 

+ 

.001 

4. J- 

3'/ 

2 

2 

- 

.002 

3 

—  2 

+ 

.021 

4 

—  2 

r 

•"'4 

5 

. o 

1 

.002 

2 

-3 

— 

.O'.J 

3 

~  3 

- 

.2  li 

4 

-3 

— 

■'  .i'> 

5 

-3 

— 

.007 

f) 

-3 

- 

.OUI 

2 

-4 

- 

.003 

3 

—  4 

— 

.014 

2w  + 

(./ 

1 

I 

+- 

.oor 

•     2 

I 

-t- 

.035 

3 

I 

+ 

.004 

I 

o 

-f- 

.002 

2 

o 

+ 

.001 

■tw  - 
4 

o 

3 

--I 

+ 

.003 

4 

—  I 

1 

5 

-1 1 

•     •  i 

.0U3 

— 

.001 

+ 

.  005 

— 

.030 

— 

•055 

— 

.027 

— 

.005 

— 

•  003 

+ 

.OK) 

+ 

2.415 

+ 

3-017 

-t- 

1 .  204 

+ 

.210 

+ 

.02() 

+ 

.  "03 

+ 

.218 

+ 

•347 

+ 

.  102 

-f- 

.031 

+ 

•  003 

-h 

.0:2 

+ 

.024 

H- 

.013 

+ 

.002 

+ 

.OUI 

+ 


4- 
+ 

-1- 


.002 
.021 

.or4 

.01)2 
.054 
.20S 
.030 
.(,07 
.001 
.003 
.UI4 

.001 

•('35 
.  004 
.002 
.001 


.003 


+ 


004 
013 
oir 
002 


001 
001 


-f- 
-t- 
+ 

-I- 
+ 

4- 
■t- 
+ 
+ 
+ 
+ 
+ 
+ 
+ 


+ 
+ 


.005 

+ 

.001 

-t- 

-f 

.001 

+ 

.001 

_ 

.001 

+ 

.005 

+ 

.001 

+ 

+ 


.002 

•  OOt 
.005 
.030 

•  "55 
.027 
.005 
.003 
.01.) 

2 .  1 1  ,j 

3.004 

'■l'J3 

•214 

.021) 

•  003 
.21.S 

•  34^ 
.161 

.031 
■  003 
.012 

.1124 
.1.13 
.002 
.ooi 


.  002 
.021 
.014 
.  002 
.054 
.2o3 
.021) 

.007 
.001 
.003 
.014 


.001  + 
.030  + 
.003     + 

.002 
.002 


.  o  I  S 

.01 1 

.002 

•  043 

.if.5 
.  005 
.005 
.  00 1 
.002 
.005 


•  03- 
.  004 


001 

002 

_u 

.002 

005 

+ 

.005 

OOI 

1 

•    ■ 

+ 

4- 
-1- 
+ 


.058 
.05S 
.020 
,001 

.00(1 

2.S16 

I.07.J 

.Ifi2 

.012 


.168  + 

.256  + 

.102  -1- 

.010  + 

.005  -f- 

.  I  >oS  + 
.003 


O.of) 

O .  of  J 

0.02 


0 

01 

2 

•  32 

n 

S.J 

t 

orj 

0 

16 

0 

01 

0 

17 

0 

26 

0 

10 

0 

01 

0 

01 

0. 

01 

o .  02 

O.lll 

o.  )4 
0.17 

o .  O I 
O.OI 


0.03 


I04 


THANSIOUMATIUN  OF  IIANSKN'S  LUNAR  THEoRy. 
'rAlll.K    W  . —  77/r    Monti's    Lllt'lllltlf — ('(intilllKHl. 


K       g 


-  3''' 


o 


sin  I  sin  (  /-fw) 


^  f.i  —  2  (.1 

2  2    I    4~ 

3  —  2  i   — 


7 
3 
4 

5 
6 

4  w  ■ 
I 


-3 
-3 
-3 
-3 
■  50' 


3  -5 

()U   —  \  (,'' 

4  -5 

5  -5 
7  (.1  —  6  (,)' 

^"  -6 


-6 
-6 
-6 


+ 
+ 
+ 
+ 
+ 


5 
6 

7 

3(j  + 


-7  j  + 

-7  I  + 

-4 


5" 

4  o 

5  o 
0  o 


.003 
.015 


004 
oSi) 

0f)0 

008 


ex)4 
004 


.001 
.001 
.001 


.005 
.002 


.031 
.060 

•043 
.014 
.002 
.004 
.010 
.008 
.001 


.001 
.001 


sin  /} 


+ 


•t- 

.031 

+ 

.060 

+ 

.043 

+ 

.014 

+ 

.002 

+ 

.004 

+ 

.010 

4- 

.ooS 

+ 

.001 

.001 

.001 


/^  -  sin  /i) 


.002 
.015 


•  !  + 
004 

O81)  I  + 

Of)0  I  — 

ooS  !  - 


004  \  + 
fx)4      — 


.001 
.001 
.001 


.005 
.002 


.not 
.no  I 

.024 

.isr, 

•  137 

.030 

.r)04 

.(«)3 

.OOl) 

.002 


— 

.004 

— 

.006 

— 

.003 



.001 

+ 

.002 

+ 

.001 

+ 

.006 

+ 

,002 

//■linen. 


ft 
/^f/illlllilV 

(I). ' 


ft 

/'><!i)uiiiiv 

(2).    ' 


■f 
4- 
+ 


+ 
+ 

4- 
+ 
+ 
+ 


+ 


+ 


002 

— 

015  1 

— 

1 

0(11 

4- 

001 

4- 

004 

4- 

o(]5 

— 

246 

- 

145 

- 

030 

— 

<X)4 

- 

003 

— 

012 

— 

(J09 

- 

oui; 

— 

001 

001 

— 

.001 

— 

005 

.002 

4- 

4- 


4- 

4- 


.031 
.060 
.043 
.014 
.002 
.004 
.010 

.ooS 
.001 


.005      — 
.(X)6  '   — 


.003  I 


.002 
.010 


,003 
.002 

,0<)2 

.odS 
.  240 

.142 
.028 
.003 
.003 
.on 
.0(i8 
.001 


.002 
.004 


.001 

on 

020 

012 
(H32 


.002 
.002   ! 


.001 

4- 

.001 

.002 

4- 

.002 

.001 

4- 

.002 

.006 

4- 

.006 

.002 

4- 

.002 

O.OI 


0.07 
0.25 
0.14 
0.03 


O.OI 

0.01 


0.02 

O.OI 


TR  \\s|.-()K\I.\ri').\  ()|-   II.WSINS  l.l'NAU    IIIIOKV. 


105 


Taum',  \'.  — 77/r   Mnuii's   I'iinilh 


il.r. 


,        D(l     )■  iTI.S  /) 

I       </  (I  -  ,  •) 


//illKill's 


l\h 


I'.MMlli.x.  I'ai.ill.ix. 


1     /Kill  II  11,1  v. \- 
•-iiii.' 
I'aiallax. 
(2) 


I),-  I)       II      I) 


0 

0 

33i;i).f)S2 

•h 

22.405 

3422.09 

3422 

•7 

3422.7 

0 

^ 

61 

3l»»-3» 

1 

0 

+ 

186.547 

— 

.  o(»4 

■t- 

186.483 

+ 

1 80 

•587 

4- 

186.55 

_ 

4 

_ 

7 

4- 

186.51 

2 

0 

+ 

10, 220 

•"59 

\ 

lo.UiI 

4- 

."o 

.  198 

4- 

10. 20 

0 

— 

4 

4- 

10.17 

3 

(J 

' 

.'.1!7 

— 

.007 

1- 

.620 

4- 

.631 

4- 

0.63 

0 

_ 

1 

4- 

0.63 

4 

0 

+ 

.041) 

•      • 

-h 

,040 

+ 

.041 

4- 

0.04 

0 

0 

4- 

0.04 

5 

u 

+ 

•  <J"3 

•(- 

.003 

4- 

.003 

,     . 

, 

J 

. 

-4 

—  1 

— 

.(XJl 

- 

.(JOI 

, 

-3 

-1 

— 

.007 

— 

.003 

— 

.010 



.oof) 



0.01 

0 

0 

—3 

-I 

— 

.067 

— 

•"55 

- 

.  122 

— 

.092 

— 

0.0(j 

0 

-(- 

3 



0. 10 

—  I 

-I 

— 

■304 

— 

.657 

- 

.  i)(i  I 

— 

.912 

— 

0.93 

.- 

2 

•t- 

3 

— 

0.95 

0 

—  1 

— 

.018 

— 

•375 

— 

•  393 

- 

■427 

— 

^•43 

0 

— 

4 

_ 

0.40 

' 

-' 

+ 

.2(J9 

+ 

.S45 

+ 

I.144 

4- 

I 

.052 

4- 

1   1 1 

4- 

6 

4- 

3 

+ 

1.16 

- 

—  I 

-f- 

.oS2 

4- 

.067 

+ 

•149 

\- 

.l"3 

4- 

0.  10 

0 

4- 

5 

f 

0.12 

3 

—  I 

+ 

.  00(J 

■h 

.003 

+ 

.UI2 

4- 

.  006 

4- 

O.OI 

0 

0 

4 

—  I 

■(- 

.001 

+ 

.  00 1 

, 

. 

. 

'.     '.   1 

—  I 

—  2 

— 

.  003 

- 

.0(17 

- 

.UK) 

— 

.010 

_ 

0.01 

0 

0 

0 

2 

- 

.ooS 

- 

.0(iS 

— 

012 

— 

O.OI 

0 

0 

,     .   ' 

• 

—  2 

+ 

,003 

+ 

.009 

+ 

.012 

+ 

.013 

4- 

O.OI 

0 

0 

1 

2 

—  2 

+ 

.001 

•      ■ 

+ 

.001 

. 

. 

. 

. 

i 

2  (J  - 

-  2w' 

i 

0 

1) 

+ 

.001 

,      . 

+ 

.001 

' 

' 

0 

•     ■ 

- 

.021 

- 

.021 

— 

.013 

— 

O.OI 

0 

4- 

,     ^ 

2 

0 

— 

.UOl 

- 

.001 

— 

.002 

.     . 

. 

; 

—  1 

—  I 

- 

.001 

- 

.(jor 

4- 

001 

,         j 

. 

0 

-1 

4- 

.010 

— 

.1)12 

— 

.002 

4- 

.  (.)( )2 

,         , 

1 

—  I 

4- 

.010 

- 

•237 



.227 

— 

379 

— 

o.?S 

0 

— 

If 

— 

0.23 

2 

-  1 

— 

.iiU 

— 

.  2S() 

- 

.301 

— 

32S 

— 

0.33 

0 

— 

— 

0.31 

3 

—  I 

— 

.<ii5 

— 

.034 

— 

."49 

— 

040 

— 

0.04 

0 

-1- 

,        J 

4 

-  1 

— 

.  (JU2 

— 

.  (X>2 

— 

.004 

— 

(->02 

,       , 

•   •! 

-3 

—  2 

— 

.()II2 

— 

,002 

,         . 

. 

1 

—  2 

-2 

- 

.018 

-t- 

.1104 

- 

.014 

~ 

.  ( >oS 

— 

0.01 

0 

0 

,      , 

—  I 

~2 

— 

.213 

+ 

.092 

— 

.  121 

— 

.  101 

__ 

0. 10 

0 

4- 

2 

— 

0.  12 

0 

-2 

— 

2.I2S 

+ 

t.S2f) 

— 

.  302 

— 

•  277 

— 

0.2S 

0 

-1 

2 

— 

0.31 

I 

-•J 

- 

.(Ji)2 

+ 

3?-3'>i 

+- 

34.301) 

+ 

34 

.  IM. 

+ 

34.2'| 

-(- 

12 

4- 

4- 

34  •  3') 

2 

—  2 

(- 

I.()(JI> 

+ 

2(1.  23^ 

-t- 

2S.225 

4- 

2S 

'70 

4-' 

2S.2(J 

4- 

n 

--i- 

4- 

2S.23 

3 

—  2 

+ 

I  .  I(|i) 

,.j_ 

■  .S.)4 

-+- 

3^'iS4 

4- 

3 

.o('i4 

4- 

3.07 

■)- 

I 

4- 

4- 

3-"') 

4 

—  2 

-t- 

•154 

-r 

.129 

f- 

•  2S3 

-(- 

271 

+ 

0.27 

0 

4- 

4- 

0.2S 

5 

-2 

+ 

.01; 

-h 

.«)S 

4- 

.023 

4- 

oiS 

+ 

0.02 

0 

•      •   1 

6 

-2 

+ 

.001 

+ 

.(X>I 

,             , 

,      .   ' 

—  2 

—  3 

- 

.001 

— 

.001 

, 

, 

,      , 

—  I 

-3 

- 

.ooS 

-f- 

.  1)04 

- 

.  004 

— 

003 

*   i 

0 

—  3 

— 

.094 

+ 

.075 

— 

.01l| 

— 

013 

— 

O.OI 

0 

4- 

1 

I 

-3 

— 

.  oOc) 

+ 

1 .  5  1 6 

+ 

I--I47 

+ 

I 

452 

4- 

1.47 

+ 

2 

- 

4- 

1-45 

2 

-3 

-1- 

.Of)t 

+ 

I.?29 

-\- 

1.920 

4- 

I 

S7f, 

4- 

1.91 

+ 

3 

4- 

1.92 

3 

-3 

+ 

.0S2 

+ 

•'47 

-f 

.229 

4- 

'97 

4- 

0.22 

+ 

2 

-r 

0.22 

4 

—  3 

+ 

.012 

+ 

.010 

+ 

.022 

^- 

Ot2 

+ 

O.OI 

0 

4- 

. 

5 

-3 

+ 

.  00 1 

+ 

.  00 1 

i 

Aifamt' 

siiiu 
Paiallax. 


io6 


TK  w'^i-mni  \  rid.'   dt  ii  \nsi;n's  i.i'nar  iiiroKV. 

'rAMM',     \ /  /"      M'ini/s     I'lll'l/ldl — ( 'cilltiinicd 


IXt 


•  con  /) 


a.w  - 

3M 

i> 

-» 

1 

-4 

2 

■4 

i 

-  » 

-1 

-4 

I 

-5 

2 

-i 

!'■'    - 

.4'.! 

•J 

-  3 

3 

-3 

4 

-  3 

—  1 

3 

-■4 

5 

-4 

J 

—  5 

3 

—  5 

4 

—  5 

—  (> 


.fU    -!•') 


3      + 

3 

3      - 


M)| 

on  ^ 

"M 
(111  I 


I'.ll. 


IlilHSili  s 

Sillf 

I'.n.ill.ix. 


IK>3 

— 

<.<3 

+ 

DillJ 

•»- 

IK.S 

t- 

mil 

+ 

oo| 

■t- 

.     . 

— 

.004 

— 

- 

.■Hi|, 

- 

.  ( ><>  1 

- 

.I.I1J 

-t* 

.1)111 

-t- 

.IJOf 

t- 

•  377 

1- 

.1122 

1- 

.577 

-1- 

(i.dJ 

s- 

.231 

^■ 

.1)12 

+ 

•  "31 

+ 

(ii)'J 

h 

.(102 

■+ 

.nol 

+ 

•"33 

H- 

(1(>2 

•h 

■  Of)  7 

+ 

""4 

.  03 1 

■+ 

III  '2 

'■ 

.1.1,5 

■(- 

. 

f- 

.i'u2 

-t. 

- 

.1104 

+ 

.  1 II  c 

+ 

.       .         t 

.  nil  1 

+ 

.oi).4 

.       ..       -t- 

.1)111 

t' 

.nil) 

1- 

.01,7 

(• 

.i'ii7 

.1102 

t 

.  1 11  )2 

.       .        -'r 

.IXJl 

+ 

.'.ml 

.       .        4- 

.(KJl 

+ 

.  00  r 

/) 

■'illlll.iv'j 

/KAiiiiiiiy 

SIIU' 

»int' 

1 

;ii.ill:ix. 

l'.ii:ill;iN 

(11 

(-'1 

OI)l 

"4') 

+ 

0()3 

T 

01  a 

+ 

oil  I 

Dill 

(1(1) 

004 

_ 

11(11  J 

— 

tl(l| 

- 

IMiS 

+ 

373 

•H 

S<)<) 

+ 

2fll 

•+- 

"43 

+ 

004 

032 

4- 

Of(^ 

+ 

"35 

•r 

01)7 

(,02 

1)114 

1J02 

1),  I    II      I)  xinc 

l'.ii,ill.i\. 


"45 

+- 

i).i)4 

1) 

t 

1 

)• 

(i.o; 

07(1 

■f 

0.  In 

■  (--  , 

3 

- 

1 

H- 

O.OI) 

(K)5 

4- 

o.UI 

u 

0 

. 

007 

- 

(joS 

— 

(  (12 

(104 

310 

t- 

4'W 

t 

lijfi 

•t- 

01  (J 

+ 

i)iti 

+ 

030 

h 

(111 

4- 

(J')4 

- 

.i'"3 

— 

.(11)2 

o(;7 

- 

.007 

- 

.007 

IJOI 

— 

.002 

— 

.""3 

0.01 

(l.OI 


0.31 

().  to 
o.  20 
(1.1), 

0.0 

"•" 
0.0 


4-  (1  f- 

1)       h  10  I- 

'<        t  (1  I 

Oil-  a 


1 

. 

_ 

.003 

— 

.003 

— 

()()2 

. 

1 

4- 

.  ()( 1 1 

-(- 

.001 

f 

002 

. 

0 

-t- 

(1(»2 

- 

.0(i2 

• 

0 

.                     . 

0 

+ 

(,3s 

- 

.038 

0 

, 

.        . 

0 

- 

•  ;<■"} 

- 

■  7"') 

- 

:oS5    - 

0.71                    <1 

0 

- 

039 

T- 

.-127 

- 

.U12 

— 

(HllJ         — 

0.01                    0 

II 

— 

002 

-;- 

(;ii2 

(1 

1 

+ 

,  002 

•)- 

.002 

■+- 

1102 

.        . 

I 

. 

+ 

.UOI 

+ 

,tK)l 

+ 

(X'2 

.       *                     • 

37 
(ill 
2(1 


i>(, 


0.7 


I  U  W'^l  |>U\I.\  I  |((\   (,|    IIWSKNS  I.INAU    llll.oKV. 
IaIII.K    \. —  'I'll,     Muiiii'a    J'didlliU ColllillllL'd. 


107 


I>(l_+  teas/) 


IVrl. 


0 

— 

1 
I 

1 

+ 

4,., 

-  a  u' 

—  2 

—  2 

- 

—  2 

- 

2  t.* 

-4.,, 

t        1 

-4 

-» 

u 

—  w' 

-1 

I) 

- 

0 

— 

1) 

-t 

(J 

-(- 

—  2 

—  1 

-(■ 

-  -I 

—  1 

H 

—  1 

1- 

■  1 

— 

—  I 

- 

-  1 
-2 

2 

—  2 

,3'.- 

-  3 '"' 

'      2 

—  2 

3 

I 

-3 

2 

-3 

— 

3 

-3 

- 

2 

--1 

!    ' 

-4 

<.i  ■( 

r<j 

I 

> 

a  — 

Ut  1 

I 

-3 

1 
1 

I  Inn  ten'  < 

siiii- 
I'.iialliii. 


. 

— 

.<«>4 

._ 

,)3S 

.(Ad 

.. 

(Hi; 

.  1  ]  J 

_. 

UJI) 

— 

•  <M7 

-_ 

001) 

— 

.in.>:^ 

— 

mil 

.       . 

_ 

CHII 

t- 

.  <  inl 

t- 

.  (  )i  }  1 

■♦- 

001 

•        •     , 

•H 

, 

-^ 

.oil 

_ 

<IOI 

— 

.(JKJ 

_ 

-,• 

.0(11 

+ 

•      • 

+ 

.{J(J2 

f- 

(Kit 

__ 

.[)(J2 

ij(jS 

+ 

.11(17 

— 

f 

.I4f- 

-t- 

(jdS 

r 

.(iiiS 

-+- 

()(ii 

,      , 

■t- 

(JDI 

, 

+- 

012 

-(• 

.0(13 

1- 

"55 

- 

■'Ml 

•t- 

0114 

— 

,(i(.) 

- 

"35 

— 

.();i 

- 

007 

— 

.(KIJ 

_ 

(JOI 

— 

- 

.()()| 

— 

— 

.  (JO  I 

— 

+ 

.  ( 1(  13 

■1- 

-1- 

.002 

H 

— 

.  o( )(} 

— 

UDI 

— 

.036 

— 

(I03 

+ 

■  00; 

-I- 

+ 

.002 
.  00 1 

+ 

.007 


.001 

.o4-i 
.  10: 
.11S3 

.00() 
.(Mil 

I) 
.  1  II  1 1 

.uul 

.1114 
.01 1 
.oul 

.  ( 10 1 

.(J02 


.003 

.  (JO  I 

.Mfi 

.Old 

.  00 1 
.  00 1 

.015 

.oil 

•'J?3 
.lull 

.0111 

.  0(  I  [ 

.004 
.001 


•  01 13 
.  1 11  )2 
.001^ 
■"37 
■ "'  '3 
.002 
.oor 


.007 


/K/.inwi/s  Pil.iniiay 

'<ini'  siiii' 

l'.irall;ix.  I'ar.ill.ix. 

(It  Vi) 


„ 

it 

,cx)6 

- 

o.ni 

.050 

— 

(I.IK 

.  !  1 17 

— 

0.  1  1 

.o'^a 

— 

u.iiS 

.IKlS 

— 

o.or 

.(i(ji 

;  ; 

.0"l 

. 

.UOI 

.(X)3 
.004 


.151       4- 
."I3        f- 


(X)7 

+ 

uo3 

+ 

938 

- 

<'V7 

— 

ood 

- 

0U2 

O'JI 

003 

002 

005 

020 

— 

old 

+ 

.007     ;     + 


I).   -I),         II 


U.OI 

O.OI 


0.15 

11.01 


U.OI 

O.OI 

"■<)1    + 


o.  10 
o.m 


0.02 
0.02 


o.ul 


Aitiimi' 

•iillU 

i'.ir.illax. 


o,  II 
O.O'J 


+ 


t'         0.14 


0.01 

O.IJ5 
>).  II 


Q^ 


< ) 


